| Russian Mathematical Surveys |
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On integrability of the deformed Ruijsenaars–Schneider system A. V. Zabrodinabc a Skolkovo Institute of Science and Technology b National Research University Higher School of Economics c National Research Centre "Kurchatov Institute"
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     1. IntroductionIntegrable many-body systems of classical mechanics play a significant role in modern mathematical physics. They are interesting and meaningful from both mathematical and physical points of view and have important applications to and deep connections with different problems in mathematics and physics. The history of integrable many-body systems starts from the famous Calogero–Moser (CM) model [1]–[4], which exists in a rational, a trigonometric (or hyperbolic), and an elliptic versions. In the most general elliptic case the equations of motion for the $N$-body CM system are
$$
\begin{equation}
\ddot x_i=4\sum_{j\ne i}^N\wp '(x_{ij}), \qquad x_{ij}=x_i-x_j,
\end{equation}
\tag{1.1}
$$
where dot denotes the time derivative. Throughout the paper, we use the standard Weierstrass $\sigma$-function, as well as the Weierstrass functions
$$
\begin{equation*}
\zeta(x)=\frac{\sigma'(x)}{\sigma(x)}\quad\text{and}\quad \wp (x)=-\zeta'(x)
\end{equation*}
\notag
$$
(see § 10 for their definition and properties). By degenerating elliptic functions to trigonometric and rational ones, one obtains the trigonometric and rational versions of the CM model. The elliptic CM model is Hamiltonian and completely integrable, that is, it has $N$ independent integrals of motion in involution. The integrability of this model was proved by different methods in [5] and [6]; also see the book [7]. It was discovered subsequently [8], [9] that there exists a one-parametric deformation of the CM system preserving integrability, often referred to as the relativistic extension. In this interpretation the parameter $\eta$ of the deformation is the inverse velocity of light. This model is now called the Ruijsenaars–Schneider (RS) system. Again, in its most general version the interaction between particles is described by elliptic functions. The equations of motion are
$$
\begin{equation}
\ddot x_i+\sum_{j\ne i}^N \dot x_i \dot x_j\bigl(\zeta (x_{ij}+\eta)+ \zeta (x_{ij}-\eta)-2\zeta(x_{ij})\bigr)=0.
\end{equation}
\tag{1.2}
$$
A properly taken limit as $\eta \to 0$ leads to equations (1.1). The RS system is Hamiltonian with Hamiltonian
$$
\begin{equation}
\mathsf{H}_1=\sum_{i=1}^N e^{p_i}\prod_{j\ne i}^N \frac{\sigma(x_{ij}+\eta)}{\sigma(x_{ij})}\,.
\end{equation}
\tag{1.3}
$$
The integrability of the RS system was proved in [9]. It conserves the quantities $\mathsf{H}_k$ and $\overline{\mathsf{H}}_k$, $k\in \mathbb{N}$, which are higher Hamiltonians in involution (for the $N$-particle system the first $N$ of them are independent). Since the seminal works [10]–[13] it became a common knowledge that integrable many-body systems of Calogero–Moser type describe the dynamics of poles of singular (generally speaking, elliptic) solutions of nonlinear integrable differential equations such as the Korteveg–de Vries (KdV) and Kadomtsev–Petviashvili (KP) ones. It was shown in [14] that the RS system plays the same role for singular solutions of the Toda lattice equation, which can be thought of as an integrable difference deformation of the KP equation. (On the Toda lattice side, the parameter $\eta$ can be identified with the lattice spacing.) Namely, the evolution of poles with respect to the time $t=t_1$ of the Toda hierarchy coincides with the RS dynamics in accordance with the equations of motion (1.2). Subsequently, this correspondence was extended [15] to the level of hierarchies: the evolution of poles in the higher times $t_k$ and $\bar t_k$ of the Toda hierarchy was shown to be described by the RS Hamiltonian flows with higher Hamiltonians $\mathsf{H}_k$ and $\overline{\mathsf{H}}_k$. Recently, a further deformation of the RS model was introduced [16] as a dynamical system describing the time evolution of poles of elliptic solutions to the Toda lattice with constraint of type B [17]. The equations of motion of the deformed RS system are
$$
\begin{equation}
\ddot x_i +\sum_{j\ne i}^N\dot x_i \dot x_j \bigl(\zeta (x_{ij}+\eta)+ \zeta (x_{ij}-\eta)-2\zeta (x_{ij})\bigr)+g(U_i^--U_i^+)=0,
\end{equation}
\tag{1.4}
$$
where
$$
\begin{equation}
U_i^{\pm}=\prod_{j\ne i}^N U^{\pm}(x_{ij}),\qquad U^{\pm}(x_{ij})= \frac{\sigma(x_{ij}\pm 2\eta) \sigma(x_{ij}\mp \eta)} {\sigma(x_{ij}\pm \eta)\sigma(x_{ij})}\,,
\end{equation}
\tag{1.5}
$$
and $g$ is the deformation parameter. For $g=0$ we have the RS system. It is obvious that $g\ne 0$ can be eliminated from the formulae by rescaling the time variable $t\to g^{-1/2}t$. In what follows we fix $g$ to be $g=\sigma(2\eta)$ without loss of generality. For this choice of $g$ equations (1.4) are exactly the same as the dynamical equations for poles under the convention on the choice of the time variable which is adopted for the Toda lattice with constraint of type B. It was shown in [16] that the limit of equations (1.4) as $\eta \to 0$ reproduces the equations of motion
$$
\begin{equation}
\ddot x_i+6\sum_{j\ne i}^N(\dot x_i+\dot x_j)\wp '(x_{ij})- 72 \sum_{j,k\ne i, j\ne k} \wp (x_{ij})\wp '(x_{ik})=0,
\end{equation}
\tag{1.6}
$$
obtained in [18] for the dynamics of the poles of elliptic solutions of the B-version of the KP equation (BKP). It was also shown in [16] that system (1.4) can be obtained by restricting the Hamiltonian flow with Hamiltonian $\mathsf{H}_1^- =\mathsf{H}_1-\overline{\mathsf{H}}_1$ of the $N=2N_0$-particle RS system to the half-dimensional subspace $\mathcal{P}\subset\mathcal{F}$ of the $4N_0$-dimensional phase space $\mathcal{F}$ corresponding to the configurations where the $2N_0$ particles stick together by grouping into $N_0$ pairs such that the distance between the particles in each pair is equal to $\eta$. Such configurations are immediately destroyed by the flow with Hamiltonian $\mathsf{H}_1^+ =\mathsf{H}_1+\overline{\mathsf{H}}_1$ but are preserved by the flow with Hamiltonian $\mathsf{H}_1^- = \mathsf{H}_1-\overline{\mathsf{H}}_1$, and the corresponding dynamics can be restricted to the subspace $\mathcal{P}$. The restriction gives equations (1.4), where $N$ is substituted by $N_0$, and $x_i$ ($i=1, \dots, N_0$) is the coordinate of the $i$th pair moving as a single whole with the fixed distance between the two particles. In fact the subspace $\mathcal{P}$ is Lagrangian; the meaning of this fact for the theory of the deformed RS system is still to be understood. In this paper we provide some evidence of the integrability of the deformed RS system (1.4). Namely, we obtain a complete set of independent integrals of motion in an explicit form. Our method is based on the fact (proved in this paper) that the subspace $\mathcal{P}$ is preserved not only by the flow with Hamiltonian $\mathsf{H}_1^-$ but also by all higher Hamiltonian flows with Hamiltonians $\mathsf{H}_k^-$. (However, the flows with Hamiltonians $\mathsf{H}_k^+$ do not preserve $\mathcal{P}$.) This offers an opportunity to obtain integrals of motion of the $N_0$-particle deformed RS system by restricting known integrals of motion for the $2N_0$-particle RS system to the subspace $\mathcal{P}$ of pairs, and this is what we do in our paper. The main result of this paper is the following explicit expressions for integrals of motion of the system (1.4) (for $g=\sigma(2\eta)$):
$$
\begin{equation}
\begin{aligned} \, \nonumber J_n&=\frac{1}{2}\sum_{m=0}^{[n/2]} \frac{\sigma(n\eta) \sigma^{2m-n}(\eta)}{m! \, (n-2m)!} \sum_{[i_1\dots i_{n-m}]}^N \dot x_{i_{m+1}}\cdots \dot x_{i_{n-m}} \prod_{\substack{\alpha,\beta =m+1 \\ \alpha <\beta}}^{n-m} V(x_{i_\alpha i_\beta }) \\ &\qquad\times \biggl[\,\prod_{\gamma =1}^m\, \prod_{\ell \ne i_1, \dots,i_{n-m}}^N U^{+}(x_{i_{\gamma}\ell})+ \prod_{\gamma =1}^m\,\prod_{\ell \ne i_1, \dots,i_{n-m}}^N U^{-}(x_{i_{\gamma}\ell})\biggr], \end{aligned}
\end{equation}
\tag{1.7}
$$
where
$$
\begin{equation*}
V(x_{ij})=\frac{\sigma^2(x_{ij})}{\sigma(x_{ij} +\eta)\sigma(x_{ij}-\eta)}
\end{equation*}
\notag
$$
and $U^{\pm}(x_{ij})$ is given in (1.5). In (1.7) $n=1,\dots, N$ and $\displaystyle{ \sum_{[i_1 \dots i_{n-m}]}^N}$ means summation over all distinct indices $i_1,\dots,i_{n-m}$ from $1$ to $N$; $[n/2]$ is the integer part of $n/2$. For $m=0$ the product $\prod_{\gamma=1}^0$ in the second line of (1.7) should be put equal to $1$. Similarly, for $2m=n$ the product $\dot x_{i_{m+1}}\cdots \dot x_{i_{n-m}}$ should also be put equal to $1$. Here are some examples for small values of $n$:
$$
\begin{equation}
\begin{aligned} \, J_1&=\sum_{i=1} \dot x_i, \\ J_2&=\frac{\sigma(2\eta)}{2\sigma^2(\eta)}\biggl[\,\sum_{i\ne j} \dot x_i \dot x_j V(x_{ij})+\sigma^2(\eta) \sum_i \biggl(\,\prod_{\ell\ne i}U^+(x_{i\ell})+ \prod_{\ell\ne i}U^-(x_{i\ell})\biggr)\biggr], \\ J_3&=\frac{\sigma(3\eta)}{6\sigma^3(\eta)} \biggl[\,\sum_{i\ne j,k, \, j\ne k} \dot x_i \dot x_j \dot x_k V(x_{ij})V(x_{ik})V(x_{jk}) \\ &\qquad+3\sigma^2(\eta) \sum_{i\ne j} \dot x_j \biggl(\,\prod_{\ell\ne i,j}U^+(x_{i\ell})+ \prod_{\ell\ne i,j}U^-(x_{i\ell})\biggr)\biggr]. \end{aligned}
\end{equation}
\tag{1.8}
$$
Note that the $m=0$ term in (1.7) is the $n$th integral of motion of the RS system (1.2). We also find the generating function of the integrals of motion
$$
\begin{equation}
R(z, u )=\det_{1\leqslant i,j\leqslant N} \bigl(z\delta_{ij}- \dot x_i \phi(x_{ij}-\eta,u)- \sigma(2\eta)z^{-1} U_i^-\phi(x_{ij}-2\eta,u)\bigr),
\end{equation}
\tag{1.9}
$$
where
$$
\begin{equation}
\phi(x,u):=\frac{\sigma(x+u)}{\sigma(u)\sigma(x)}\,.
\end{equation}
\tag{1.10}
$$
The equation $R(z,u)=0$ defines the spectral curve, which is an integral of motion. An interesting problem of integrable time discretization of the CM and RS systems was addressed in [19] and [20]; also see the book [21]. This problem is closely connected with the Bäcklund transformations of the CM and RS systems [22]–[24] and the so-called self-dual form of their equations of motion [25], [26]. The idea is that the Bäcklund transformation can be regarded as one-step evolution in discrete time. The equations of motion of the most general elliptic version of the $N$-particle RS system in discrete time obtained in [20] are as follows:
$$
\begin{equation}
\begin{aligned} \, \nonumber &\prod_{k=1}^N \sigma(x_i^{n}-x_k^{n+1}-\eta) \sigma(x_i^{n}-x_k^{n}+\eta) \sigma(x_i^{n}-x_k^{n-1}) \\ &\qquad+\prod_{k=1}^N\sigma(x_i^{n}-x_k^{n+1})\sigma(x_i^{n}-x_k^{n}-\eta) \sigma(x_i^{n}-x_k^{n-1}+\eta)=0, \end{aligned}
\end{equation}
\tag{1.11}
$$
where $x_i^n$ is the coordinate of the $i$th particle at the $n$th step of discrete time. The continuous-time limit of (1.11) taken properly yields the equations of motion of the RS model (1.2). In this paper we propose an integrable time discretization of the deformed RS system obtained in a way similar to the one which derives (1.11) from (1.2). The equations of motion in discrete time are
$$
\begin{equation}
\begin{aligned} \, \nonumber &\mu \prod_{k=1}^N \sigma(x_i^{n}-x_k^{n+1}) \sigma(x_i^{n}- x_k^{n}+\eta) \sigma(x_i^{n}-x_k^{n-1}-\eta) \\ \nonumber &\qquad\qquad+\mu \prod_{k=1}^N \sigma(x_i^{n}-x_k^{n+1}+\eta) \sigma(x_i^{n}-x_k^{n}-\eta)\sigma(x_i^{n}-x_k^{n-1}) \\ \nonumber &\qquad=\mu^{-1} \prod_{k=1}^N \sigma(x_i^{n}-x_k^{n+1}-\eta) \sigma(x_i^{n}-x_k^{n}+\eta) \sigma(x_i^{n}-x_k^{n-1}) \\ &\qquad\qquad+\mu^{-1}\prod_{k=1}^N \sigma(x_i^{n}-x_k^{n+1}) \sigma(x_i^{n}-x_k^{n}-\eta)\sigma(x_i^{n}-x_k^{n-1}+\eta), \end{aligned}
\end{equation}
\tag{1.12}
$$
where $\mu$ is a parameter related to the lattice spacing in the time lattice. The structure of each side is the same as the structure of the left-hand side of (1.11). We note that this form of equations of motion was conjectured by Krichever some time ago. There are two different continuous-time limits of equations (1.12). One of them gives the equations of motion of the deformed RS system (1.4). The other leads to the equations obtained in [18] as the equations of motion of poles of elliptic solutions of the semi-discrete BKP equation. Like the equations of motion for the RS system, equations (1.12) admit trigonometric (or hyperbolic) and rational degenerations. In the trigonometric limit one of the two quasi-periods of the $\sigma$-function tends to $\infty$ and $\sigma(x)\to \sin x$. In the rational limit both quasi-periods tend to $\infty$ and $\sigma(x)\to x$. In this paper we do not discuss the specifics and details of the trigonometric and rational limits. We also present the extension of the deformed RS model to a lattice field theory (a ‘field analogue’) in which the coordinates of particles $x_i$ become ‘fields’ $x_i(x,t)$, which depend not only on time $t$ but also on the space variables $x$. Following the method developed in [27] for the CM model and applied in [28] to the RS model, we obtain the equations of motion as equations for poles of more general elliptic solutions (called elliptic families in [27]) of the fully discrete BKP equation. The organization of the paper is as follows. In § 2 we recall the main facts about the elliptic RS model. In § 3 we show, by reproducing the result of [16], that the dynamics of the deformed RS system is the $\mathsf{H}_1^{-}$-flow of the RS system restricted to the space of pairs. The core of the paper is § 4, where we prove that the space of pairs is invariant under all higher $\mathsf{H}^{-}_k$-flows and find integrals of motion of the deformed RS system in the explicit form, as well as the generating function of them. In § 5 we obtain the Bäcklund transformation of the deformed RS system. In § 6 the chain of Bäcklund transformations is interpreted as discrete-time evolution, and the equations of motion in discrete time are obtained. Possible continuum limits are also discussed. In § 7 we show that the discrete-time equations of motion for the deformed RS system (1.12) describe the dynamics of poles of elliptic solutions of the fully discrete BKP equation. Section 8 is devoted to obtaining the lattice field analogue of the fully discrete deformed RS system. In § 9 we make concluding remarks and list some open problems. There also are two appendices. In Appendix A (§ 10) the definition and main properties of the Weierstrass functions are presented. In Appendix B (§ 11) we prove an identity for elliptic functions which is the key identity for the proof of Theorem 4.1 in § 4. This paper has grown up from our joint works [16] and [17] with Igor Krichever. Soon after the present work had started, my older friend and co-author Igor Krichever passed away. He worked till the last of his days, and we had several illuminating conversations. With sorrow and gratefulness, I dedicate this paper to his memory. 2. The RS systemHere we collect the main facts on the elliptic RS system by following [9]. The $N$-particle elliptic RS model is a completely integrable Hamiltonian system. The canonical Poisson brackets between coordinates and momenta are $\{x_i, p_j\}=\delta_{ij}$. The integrals of motion in involution have the form
$$
\begin{equation}
\mathsf{I}_n=\sum_{\mathcal{I}\subset \{1,\dots,N\}, \, |\mathcal{I}|=n} \exp \biggl(\,\sum_{i\in \mathcal{I}}p_i\biggr) \prod_{i\in \mathcal{I},\, j\notin \mathcal{I}} \frac{\sigma(x_{ij}+\eta)}{\sigma(x_{ij})}\,, \qquad n=1,\dots,N.
\end{equation}
\tag{2.1}
$$
The sum is taken over all subsets $\mathcal{I}$ of the set $\{1,\dots,N\}$ with $n$ elements. It is natural to put $\mathsf{I}_0=1$. Important particular cases of (2.1) are
$$
\begin{equation}
\mathsf{I}_1=\sum_{i=1}^N e^{p_i}\prod_{j\ne i} \frac{\sigma(x_{ij}+\eta)}{\sigma(x_{ij})}\,,
\end{equation}
\tag{2.2}
$$
which is the Hamiltonian $\mathsf{H}_1$ of the chiral RS model, and Compared to [9], our formulae differ by the canonical transformation
$$
\begin{equation*}
e^{p_i}\to e^{p_i}\prod_{j\ne i} \frac{\sigma^{1/2}(x_{ij}+\eta)}{\sigma^{1/2} (x_{ij}-\eta)}\,,\quad x_i\to x_i,
\end{equation*}
\notag
$$
which allows one to eliminate square roots from the formulae in [9]. Let us denote the time variable of the Hamiltonian flow with Hamiltonian $\mathsf{H}_1=\mathsf{I}_1$ by $t_1$. The velocities of the particles are
$$
\begin{equation}
\overset{*}x_i=\frac{\partial \mathsf{H}_1}{\partial p_i}= e^{p_i}\prod_{j\ne i}\frac{\sigma(x_{ij}+\eta)}{\sigma(x_{ij})}\,,
\end{equation}
\tag{2.4}
$$
where asterisk means the $t_1$-derivative. Note that in terms of velocities the integrals of motion (2.1) read
$$
\begin{equation}
\mathsf{I}_n=\frac{1}{n!} \sum_{[i_1 \dots i_n]}^N \overset{*}x_{i_1}\cdots \overset{*}x_{i_n} \prod_{\substack{\alpha,\beta =1 \\ \alpha <\beta}}^{n} \frac{\sigma^2(x_{i_\alpha i_\beta})} {\sigma(x_{i_\alpha i_\beta}+\eta)\sigma(x_{i_\alpha i_\beta}-\eta)}\,.
\end{equation}
\tag{2.5}
$$
Here $\sum_{[i_1 \dots i_{n}]}^N$ means summation over all distinct indices $i_1, \dots , i_{n}$ from $1$ to $N$. It is not difficult to verify that the Hamiltonian equations $\overset{*}p_i=-\partial \mathsf{H}_1/\partial x_i$ are equivalent to the following equations of motion:
$$
\begin{equation}
\overset{**}x_i+\sum_{k\ne i}^N\overset{*}x_i\overset{*}x_k \bigl(\zeta(x_{ik}+\eta)+\zeta(x_{ik}-\eta)-2\zeta(x_{ik})\bigr)=0,
\end{equation}
\tag{2.6}
$$
which are equations (1.2). One can also introduce integrals of motion $\mathsf{I}_{-n}$ as
$$
\begin{equation}
\mathsf{I}_{-n}=\mathsf{I}_{N}^{-1}\mathsf{I}_{N-n}= \sum_{\mathcal{I}\subset \{1,\dots,N\}, \, |\mathcal{I}|=n} \exp\biggl(-\sum_{i\in \mathcal{I}}p_i\biggr) \prod_{i\in \mathcal{I},\, j\notin \mathcal{I}} \frac{\sigma(x_{ij}-\eta)}{\sigma(x_{ij})}\,.
\end{equation}
\tag{2.7}
$$
In particular,
$$
\begin{equation}
\mathsf{I}_{-1}=\sum_{i=1}^N e^{-p_i}\prod_{j\ne i} \frac{\sigma(x_{ij}-\eta)}{\sigma(x_{ij})}\,.
\end{equation}
\tag{2.8}
$$
It can easily be verified that the equations of motion with respect to the time $\bar t_1$ corresponding to the Hamiltonian $\overline{\mathsf{H}}_1=\sigma^2(\eta)\mathsf{I}_{-1}$ are the same as (1.2) (and (2.6)). We introduce renormalized integrals of motion:
$$
\begin{equation}
\mathsf{J}_n=\frac{\sigma(|n|\eta)}{\sigma^n(\eta)}\, \mathsf{I}_n,\qquad n=\pm 1,\dots,\pm N.
\end{equation}
\tag{2.9}
$$
It was shown in [15] that the higher Hamiltonians of the RS model can be obtained from the equation of the spectral curve
$$
\begin{equation}
z^N +\sum_{n=1}^N \phi_n(\lambda )\, \mathsf{J}_n \, z^{N-n}=0, \quad \phi_n(\lambda)=\frac{\sigma(\lambda-n\eta)}{\sigma(\lambda)\sigma(n\eta)}\,,
\end{equation}
\tag{2.10}
$$
as
$$
\begin{equation}
\mathsf{H}_n=\operatorname*{res}_{z=\infty} \bigl(z^{n-1}\lambda (z)\bigr).
\end{equation}
\tag{2.11}
$$
In general, they are expressed as
$$
\begin{equation}
\begin{aligned} \, \mathsf{H}_n&=\mathsf{J}_n +Q_n(\mathsf{J}_1,\dots,\mathsf{J}_{n-1}), \\ \overline{\mathsf{H}}_n&=\mathsf{J}_{-n} + Q_n(\mathsf{J}_{-1},\dots,\mathsf{J}_{-n+1}) \end{aligned}
\end{equation}
\tag{2.12}
$$
for $n\in \mathbb{N}$, where $Q_n$ are some homogeneous polynomials with degree of homogeneity $n$ (where the degree of $\mathsf{J}_k$ is set equal to $k$). For example,
$$
\begin{equation}
\begin{aligned} \, \mathsf{H}_1&=\mathsf{J}_1, \\ \mathsf{H}_2&=\mathsf{J}_2 -\zeta (\eta)\mathsf{J}_1^2, \\ \mathsf{H}_3&=\mathsf{J}_3 -(\zeta (\eta)+ \zeta (2\eta))\mathsf{J}_1\mathsf{J}_2 + \biggl(\frac{3}{2}\, \zeta^2(\eta)- \frac{1}{2}\,\wp (\eta)\biggr)\mathsf{J}_1^3 \end{aligned}
\end{equation}
\tag{2.13}
$$
(see [15]). We also introduce the Hamiltonians
$$
\begin{equation}
\mathsf{H}_n^{\pm}=\mathsf{H}_n \pm \overline{\mathsf{H}}_n.
\end{equation}
\tag{2.14}
$$
On the Toda lattice side, the RS dynamics corresponds to the dynamics of poles of elliptic solutions and the Hamiltonians $\mathsf{H}_n^{\pm}$ generate the flows $\partial_{t_n}\pm \partial_{\bar t_n}$, where $t_n$ and $\bar t_n$ are canonical higher times of the Toda lattice hierarchy.
3. The deformed RS model as a dynamical system for pairs of RS particlesIn this section we reproduce the result of [16] and show that the restriction of the RS dynamics of $N=2N_0$ particles to the subspace $\mathcal{P}$ in which the particles stick together in $N_0$ pairs such that (see Fig. 1), leads to the equations of motion of the deformed RS system for the coordinates of pairs. It is natural to introduce the variables which are the coordinates of pairs. It was proved in [16] that such a structure is preserved by the $\mathsf{H}_1^-$-flow $\partial_t=\partial_{t_1}-\partial_{\bar t_1}$ but is destroyed by the $\mathsf{H}_1^+$-flow $\partial_{t_1}+\partial_{\bar t_1}$. Therefore, to define the dynamical system we should fix $T_1^+=(t_1+\bar t_1)/2$ to be $0$, that is, put $\bar t_1 =-t_1$, and consider the evolution with respect to the time $t=T_1^-=(t_1 -\bar t_1)/2$.For the velocities $\dot x_i =\partial \mathsf{H}_1^-/\partial p_i$ we have
$$
\begin{equation}
\dot x_{2i-1}=e^{p_{2i-1}}\prod_{j=1, \ne 2i-1}^{2N_0} \frac{\sigma(x_{2i-1, j}+\eta)}{\sigma(x_{2i-1,j})}+ \sigma^2(\eta)e^{-p_{2i-1}}\prod_{j=1, \ne 2i-1}^{2N_0} \frac{\sigma(x_{2i-1, j}-\eta)}{\sigma(x_{2i-1, j})}
\end{equation}
\tag{3.3}
$$
and
$$
\begin{equation}
\dot x_{2i}=e^{p_{2i}} \prod_{j=1,\ne2i}^{2N_0} \frac{\sigma(x_{2i, j}+\eta)}{\sigma(x_{2i,j})}+ \sigma^2(\eta)e^{-p_{2i}}\prod_{j=1, \ne 2i}^{2N_0} \frac{\sigma(x_{2i, j}-\eta)}{\sigma(x_{2i, j})}\,.
\end{equation}
\tag{3.4}
$$
Under the constraint (3.1) the first term on the right-hand side of (3.3) vanishes. The second term on the right-hand side of (3.4) also vanishes. Then in terms of the coordinates $X_i$ of pairs equations (3.3) and (3.4) read:
$$
\begin{equation}
\begin{aligned} \, \dot x_{2i-1}&=\sigma(\eta)\sigma(2\eta)e^{-p_{2i-1}}\prod_{j=1,\ne i}^{N_0} \frac{\sigma(X_{ij}-2\eta)}{\sigma(X_{ij})}\,, \\ \dot x_{2i}&=\frac{\sigma(2\eta)}{\sigma(\eta)} e^{p_{2i}} \prod_{j=1, \ne i}^{N_0}\frac{\sigma(X_{ij}+2\eta)}{\sigma(X_{ij})}\,. \end{aligned}
\end{equation}
\tag{3.5}
$$
It is clear from (3.5) that if we set
$$
\begin{equation}
p_{2i-1}=\alpha_i+P_i, \quad p_{2i}=\alpha_i-P_i, \qquad i=1,\dots,N_0,
\end{equation}
\tag{3.6}
$$
where
$$
\begin{equation}
\alpha_i= \log \sigma(\eta)+\frac{1}{2}\sum_{j\ne i}^{N_0} \log \frac{\sigma(X_{ij}-2\eta)}{\sigma(X_{ij}+2\eta)}
\end{equation}
\tag{3.7}
$$
and the $P_i$ are arbitrary, then we have $\dot x_{2i-1}= \dot x_{2i}$ for any $i$, so that the dynamics preserves the distance between the particles in each pair. Under the $\mathsf{H}_1^{-}$-flow each pair moves as a single whole. Then equations (3.5) are equivalent to the single equation
$$
\begin{equation}
\dot X_i=\sigma(2\eta)e^{-P_i}\prod_{j\ne i}^{N_0} \frac{(\sigma(X_{ij}-2\eta)\sigma(X_{ij}+2\eta))^{1/2}}{\sigma(X_{ij})}\,.
\end{equation}
\tag{3.8}
$$
We have passed from the initial $4N_0$-dimensional phase space $\mathcal{F}$ with coordinates $(\{x_i\}_N, \{p_i\}_N)$ to the $2N_0$-dimensional subspace $\mathcal{P}\subset \mathcal{F}$ of pairs defined by the constraints
$$
\begin{equation}
\begin{cases} x_{2i}-x_{2i-1}=\eta,\ x_{2i-1}=X_i, \\ p_{2i-1}+p_{2i}=2\log \sigma(\eta)+\displaystyle\sum_{j\ne i} \log \dfrac{\sigma(X_{ij}-2\eta)}{\sigma(X_{ij}+2\eta)}\,. \end{cases}
\end{equation}
\tag{3.9}
$$
The coordinates in $\mathcal{P}$ are $(\{X_i\}_{N_0}, \{P_i\}_{N_0})$. Proposition 3.1. The subspace $\mathcal{P}\subset \mathcal{F}$ defined by (3.9) is Lagrangian. Proof. We should prove that the restriction of the canonical 2-form to the half-dimensional subspace $\mathcal{P}$ is identically zero. This is a simple calculation with the help of equations (3.6), (3.7), and (3.9). $\Box$ Theorem 3.1. The subspace $\mathcal{P}$ is preserved by the Hamiltonian flow with Hamiltonian $\mathsf{H}_1^{-}=\mathsf{H}_1-\overline{\mathsf{H}}_1$, and the equations of motion of the deformed RS model (1.4) are obtained as the restriction of this flow to the subspace $\mathcal{P}$. Proof. Restricting the second set of the Hamiltonian equations,
$$
\begin{equation*}
\dot p_i=-\frac{\partial\mathsf{H}_1^-}{\partial x_i}\,,
\end{equation*}
\notag
$$
to the subspace $\mathcal{P}$ we obtain
$$
\begin{equation}
\begin{aligned} \, \nonumber &\dot p_{2i-1}=\sigma(\eta)\sigma(2\eta)e^{-\alpha_i -P_i} \\ \nonumber &\quad\times\prod_{k=1, \ne i}^{N_0} \frac{\sigma(X_{ik}-2\eta)}{\sigma(X_{ik})} \biggl[\,\sum_{j=1,\ne i}^{N_0}\bigl(\zeta(X_{ij}-2\eta)- \zeta(X_{ij})\bigr)+\zeta (\eta)-\zeta(2\eta)\biggr] \\ \nonumber &\quad+\sigma(\eta)\sigma(2\eta)\sum_{l=1,\ne i}^{N_0}e^{-\alpha_l -P_l} \prod_{k=1, \ne l}^{N_0}\frac{\sigma(X_{lk}-2\eta)}{\sigma(X_{lk})} \bigl(\zeta(X_{il}+\eta)-\zeta (X_{il})\bigr) \\ \nonumber &\quad-\frac{\sigma(2\eta)}{\sigma(\eta)}\sum_{l=1}^{N_0} e^{\alpha_l-P_l} \prod_{k=1, \ne l}^n\frac{\sigma(X_{lk}+2\eta)}{\sigma(X_{lk})} \bigl(\zeta(X_{il}-2\eta)-\zeta (X_{il}-\eta)\bigr) \\ &\quad+\sigma^{-1}(\eta)e^{\alpha_i+P_i}\prod_{k=1, \ne i}^{N_0} \frac{\sigma(X_{ik}+\eta)}{\sigma(X_{ik}-\eta)}- \sigma(\eta)e^{-\alpha_i+P_i}\prod_{k=1, \ne i}^{N_0} \frac{\sigma(X_{ik}-\eta)}{\sigma(X_{ik}+\eta)}\,. \end{aligned}
\end{equation}
\tag{3.10}
$$
Taking the time derivative of (3.8) we obtain:
$$
\begin{equation}
\begin{aligned} \, \nonumber \ddot X_i&=-\sigma(2\eta)\dot P_i e^{-P_i}\prod_{j\ne i}^{N_0} \frac{(\sigma(X_{ij}-2\eta)\sigma(X_{ij}+2\eta))^{1/2}}{\sigma(X_{ij})} \\ &\qquad+\frac{1}{2}\sum_{j\ne i}^{N_0}\dot X_i(\dot X_i-\dot X_j) \bigl(\zeta(X_{ij}-2\eta)+\zeta(X_{ij}+2\eta)-2\zeta (X_{ij})\bigr). \end{aligned}
\end{equation}
\tag{3.11}
$$
To find $P_i$, in the equality $\dot P_i=-\dot \alpha_i +\dot p_{2i-1}$ we must substitute the expression for $p_{2i-1}$ from (3.10) taking (3.8) into account:
$$
\begin{equation*}
\begin{aligned} \, \dot P_i&=-\dot \alpha_i +\dot X_i \biggl[\,\sum_{j\ne i}^{N_0} \bigl(\zeta (X_{ij}-2\eta)-\zeta (X_{ij})\bigr) +\zeta (\eta)- \zeta (2\eta)\biggr] \\ &\qquad+\sum_{l\ne i}^{N_0} \dot X_l \bigl(\zeta (X_{il}+\eta)- \zeta(X_{il})\bigr)-\sum_{l=1}^{N_0}\dot X_l\bigl(\zeta(X_{il}-2\eta)- \zeta (X_{il}-\eta)\bigr) \\ &\qquad+e^{P_i}\prod_{k\ne i}^{N_0} \frac{\sigma^{1/2}(X_{ik}-2\eta)\sigma(X_{ik}+\eta)} {\sigma^{1/2}(X_{ik}+2\eta)\sigma(X_{ik}-\eta)}-e^{P_i}\prod_{k\ne i}^{N_0} \frac{\sigma^{1/2}(X_{ik}+2\eta) \sigma(X_{ik}-\eta)} {\sigma^{1/2}(X_{ik}-2\eta)\sigma(X_{ik}+\eta)}\,. \end{aligned}
\end{equation*}
\notag
$$
Plugging here $\dot \alpha_i$ from (3.7) and substituting the result into (3.11) we finally obtain:
$$
\begin{equation}
\ddot X_i=-\sum_{j\ne i}^{N_0}\dot X_i \dot X_j\bigl(\zeta (X_{ij}+\eta)+ \zeta (X_{ij}-\eta)-2\zeta(X_{ij})\bigr)+\sigma(2\eta)(U_i^+-U_i^-),
\end{equation}
\tag{3.12}
$$
where
$$
\begin{equation}
U_i^{\pm}=\prod_{j\ne i}^{N_0} \frac{\sigma(X_{ij}\pm 2\eta)\sigma(X_{ij}\mp\eta)} {\sigma(X_{ij}\pm\eta)\sigma(X_{ij})}\,.
\end{equation}
\tag{3.13}
$$
These are equations (1.1), (1.2) of the deformed RS system (for $g=\sigma(2\eta)$ and $N=N_0$). $\Box$ 4. Integrals of motionIn this section we are going to prove that the subspace $\mathcal{P}$ is invariant not only with respect to the $\mathsf{H}_1^{-}$-flow but also with respect to all higher $\mathsf{H}_k^{-}$-flows. This offers an opportunity to obtain integrals of motion $J_n$ of the deformed RS model by restricting the RS integrals of motion $\mathsf{J}_n$ and $\mathsf{J}_{-n}$ to the subspace $\mathcal{P}$. We denote the restriction of $\mathsf{J}_k$ by $J_k$:
$$
\begin{equation}
J_k(\{X_i\}_{N_0},\{P_i\}_{N_0})= \mathsf{J}_k(\{x_\ell\}_N,\{p_\ell\}_N)\big|_{\mathcal{P}}, \qquad k\in \mathbb{Z}.
\end{equation}
\tag{4.1}
$$
The notation $\mathsf{J}_k (\{x_\ell\}_N,\{p_\ell\}_N)\big|_{\mathcal{P}}$ means that the variables $x_{\ell}$ and $p_{\ell}$ are constrained by relations (3.9), that is,
$$
\begin{equation*}
\begin{alignedat}{2} x_{2i-1}&=X_i, &\qquad x_{2i}&=X_i+\eta, \\ p_{2i-1}&=\alpha_i(\{X_j\}_{N_0})+P_i, &\qquad p_{2i}&=\alpha_i(\{X_j\}_{N_0})-P_i, \end{alignedat}
\end{equation*}
\notag
$$
where $\alpha_i$ is given by (3.7). Note that $J_k$ can be regarded as a function of $\{X_j\}_{N_0}$ and $\{\dot X_j\}_{N_0}$ by virtue of (3.8), and
$$
\begin{equation*}
\frac{\partial J_k}{\partial P_i}= -\dot X_i\,\frac{\partial J_k}{\partial \dot X_i}\,.
\end{equation*}
\notag
$$
Similar notation will be used for the restrictions of Hamiltonians:
$$
\begin{equation}
\begin{aligned} \, H_k (\{X_i\}_{N_0},\{P_i\}_{N_0})&= \mathsf{H}_k(\{x_\ell\}_N,\{p_\ell\}_N)\big|_{\mathcal{P}}, \\ \bar H_k(\{X_i\},\{P_i\})&= \overline{\mathsf{H}}_k(\{x_\ell\}_N,\{p_\ell\}_N)\big|_{\mathcal{P}}. \end{aligned}
\end{equation}
\tag{4.2}
$$
4.1. The invariance of the subspace $\mathcal{P}$ under the Hamiltonian flows $\partial_{t_k}- \partial_{\bar t_k}$This subsection is devoted to the proof of the following theorem. Theorem 4.1. The space $\mathcal{P}$ of pairs defined by (3.9) is invariant with respect to the Hamiltonian flows $\partial_{t_k}-\partial_{\bar t_k}$ with Hamiltonians $\mathsf{H}_k^{-}$ for all $k\geqslant 1$. The explicit expressions for integrals of motion of the deformed RS system will follow from the proof. To prove that the first set of constraints, $x_{2i-1}-x_{2i}=\eta$, is preserved, we have to show that
$$
\begin{equation*}
(\partial_{t_k}-\partial_{\bar t_k})x_{2i-1}= (\partial_{t_k}-\partial_{\bar t_k})x_{2i}\quad\text{for all } i=1,\dots,N_0,
\end{equation*}
\notag
$$
that is,
$$
\begin{equation}
\frac{\partial \mathsf{H}_k}{\partial p_{2i-1}}- \frac{\partial \overline{\mathsf{H}}_k}{\partial p_{2i-1}}= \frac{\partial \mathsf{H}_k}{\partial p_{2i}}- \frac{\partial \overline{\mathsf{H}}_k}{\partial p_{2i}}\,,
\end{equation}
\tag{4.3}
$$
if the coordinates and momenta are restricted to the space $\mathcal{P}$. Note that equations (3.6) imply that $\partial_{p_{2i-1}}-\partial_{p_{2i}}=\partial_{P_i}$, so (4.3) is equivalent to
$$
\begin{equation}
\frac{\partial H_k}{\partial P_i}=\frac{\partial \bar H_k}{\partial P_i}\,.
\end{equation}
\tag{4.4}
$$
It follows from (2.12) that it is enough to prove that $J_n =J_{-n}$. Let $\mathcal{N}$ be the set $\mathcal{N}=\{1, \dots , N_0\}$. Separating the summation over odd and even indices in (2.1) (where there are $m$ odd and $n-m$ even indices), for $0<n\leqslant N_0$ we can write where
$$
\begin{equation}
\begin{aligned} \, \nonumber J_{n,m}&=\frac{\sigma(n\eta)}{\sigma^n(\eta)}\, \sum_{\substack{\mathcal{I},\mathcal{J}\subseteq \mathcal{N} \\ |\mathcal{I}|=m, \,|\mathcal{J}|=n-m}}\biggl(\,\prod_{i\in \mathcal{I}} e^{p_{2i-1}}\biggr)\biggl(\,\prod_{j\in \mathcal{J}}e^{p_{2j}}\biggr) \\ \nonumber &\qquad\times \prod_{\ell \in \mathcal{N}\setminus \mathcal{I}}\, \prod_{i\in \mathcal{I}}\frac{\sigma(X_{i\ell}+\eta)}{\sigma(X_{i\ell})} \prod_{\ell \in \mathcal{N}\setminus \mathcal{I}}\, \prod_{j\in \mathcal{J}} \frac{\sigma(X_{j\ell}+2\eta)}{\sigma(X_{i\ell}+\eta)} \\ &\qquad \times \prod_{\ell \in \mathcal{N}\setminus \mathcal{J}}\, \prod_{i\in \mathcal{I}} \frac{\sigma(X_{i\ell})}{\sigma(X_{i\ell}-\eta)} \prod_{\ell \in \mathcal{N}\setminus \mathcal{J}}\, \prod_{j\in \mathcal{J}} \frac{\sigma(X_{j\ell}+\eta)}{\sigma(X_{j\ell})}. \end{aligned}
\end{equation}
\tag{4.6}
$$
Clearly, this is zero unless $\mathcal{I}\cap (\mathcal{N}\setminus \mathcal{J})=\varnothing$, that is, the set $\mathcal{I}$ should be contained in $\mathcal{J}$, $\mathcal{I}\subseteq \mathcal{J}$. Since $|\mathcal{I}|=m$ and $|\mathcal{J}|=n-m$, this is possible only for $m\leqslant [n/2]$, otherwise $J_{n,m}$ vanishes. Then using (3.6)–(3.8) we obtain
$$
\begin{equation*}
\begin{aligned} \, \biggl(\,\prod_{i\in \mathcal{I}}e^{p_{2i-1}}\biggr) \biggl(\,\prod_{j\in \mathcal{J}}e^{p_{2j}}\biggr) &=\frac{\sigma^n(\eta)}{\sigma^{n-2m}(2\eta)} \biggl(\,\prod_{i\in \mathcal{I}}\, \prod_{\ell \in \mathcal{N}\setminus \{i\}} \frac{\sigma(X_{i\ell}-2\eta)}{\sigma(X_{i\ell}+2\eta)}\biggr) \\ &\qquad\times\biggl(\,\prod_{j\in \mathcal{J}\setminus \mathcal{I}}\, \prod_{\ell \in \mathcal{N}\setminus \{j\}} \frac{\sigma(X_{j\ell})}{\sigma(X_{j\ell}+2\eta)}\biggr) \prod_{j\in \mathcal{J}\setminus \mathcal{I}}\dot X_j. \end{aligned}
\end{equation*}
\notag
$$
The expression for $J_{-n,m}$ is similar but in that case $m$ is the number of even, rather than odd, indices and $\eta$ must be replaced by $-\eta$ in all factors in the products. After plugging this into (4.6) and making cancellations, we obtain:
$$
\begin{equation}
\begin{aligned} \, \nonumber J_{\pm n,m}&=\frac{\sigma(n\eta)}{\sigma^{n-2m}(\eta)} \sum_{\substack{\mathcal{J}\\ |\mathcal{J}|=n-m}} \, \sum_{\substack{\mathcal{I}\subseteq \mathcal{J}\\ |\mathcal{I}|=m}} \biggl(\,\prod_{j\in \mathcal{J}\setminus \mathcal{I}} \dot X_j\biggr) \\ &\qquad\times\biggl(\,\prod_{\substack{i,j\in \mathcal{J}\setminus\mathcal{I} \\ i<j}} V(X_{ij})\biggr) \biggl(\,\prod_{i\in \mathcal{I}}\, \prod_{\ell \in \mathcal{N} \setminus \mathcal{J}}U^{\pm}(X_{i\ell})\biggr), \end{aligned}
\end{equation}
\tag{4.7}
$$
where
$$
\begin{equation}
V(X_{ij})=\frac{\sigma^2(X_{ij})}{\sigma(X_{ij}+\eta)\sigma(X_{ij}-\eta)}
\end{equation}
\tag{4.8}
$$
and
$$
\begin{equation}
U^{\pm}(X_{ij})=\frac{\sigma(X_{ij}\pm 2\eta)\sigma(X_{ij}\mp \eta)} {\sigma(X_{ij}\pm \eta)\sigma(X_{ij})}\,.
\end{equation}
\tag{4.9}
$$
Going over from summation over the subsets $\mathcal{J}\subset \mathcal{N}$ and $\mathcal{I}\subseteq \mathcal{J}$ to summation over the subsets $\mathcal{I}$ and $\mathcal{I}'$ such that $\mathcal{I}\cap \mathcal{I}'=\varnothing$ ($\mathcal{I}'=\mathcal{J}\setminus \mathcal{I}$), we can write the right-hand side of (4.7) in the form
$$
\begin{equation}
\begin{aligned} \, \nonumber J_{\pm n, m}&=\frac{\sigma(n\eta)}{\sigma^{n-2m}(\eta)} \sum_{\substack{\mathcal{I}, \mathcal{I}', \, \mathcal{I}\cap \mathcal{I}'=\varnothing \\ |\mathcal{I}|=m, \, |\mathcal{I}'|=n-2m}} \biggl(\,\prod_{j\in \mathcal{I}'}\dot X_j\biggr) \biggl(\,\prod_{\substack{i,j\in \mathcal{I}' \\ i<j}} V(X_{ij})\biggr) \\ &\qquad\times\biggl(\,\prod_{i\in \mathcal{I}}\, \prod_{\ell \in \mathcal{N} \setminus (\mathcal{I}\cup \mathcal{I}')} U^{\pm}(X_{i\ell})\biggr). \end{aligned}
\end{equation}
\tag{4.10}
$$
The equality $J_{n,m}=J_{-n,m}$ is a consequence of the following lemma. Lemma 4.1. For any $\mathcal{N}'\subseteq \mathcal{N}=\{1,\dots,N_0\}$ the following holds: This lemma is proved in Appendix B (§ 11). Applying this lemma for $\mathcal{N}'=\mathcal{N}\setminus \mathcal{I}'$ to (4.10) we see that $J_{n,m}=J_{-n,m}$. Formula (1.7) for the integrals of motion in the introduction is an explicitly symmetrized version of (4.10): We have proved half of the statement of Theorem 4.1, namely, that the first set of constraints in (3.9), $x_{2i}-x_{2i-1}=\eta$, is invariant under the flows $\partial_{t_k}-\partial_{\bar t_k}$.Let us prove that the second set of constraints in (3.9) is preserved too. We must show that the equality in (3.9) remains true after applying $\partial_{t_n}-\partial_{\bar t_n}$ to both sides. Then on the left-hand side we have
$$
\begin{equation*}
\frac{\partial \mathsf{H}^{-}_n}{\partial x_{2i-1}}+ \frac{\partial \mathsf{H}^{-}_n}{\partial x_{2i}}= \frac{\partial \mathsf{H}^{-}_n}{\partial X_{i}}\,.
\end{equation*}
\notag
$$
Without loss of generality we can put $i=1$ for the simplicity of notation. Then we have to prove that
$$
\begin{equation*}
\frac{\partial \mathsf{H}^{-}_n}{\partial X_1}= \sum_{k\ne 1} \biggl(\frac{\partial\mathsf{H}^{-}_n}{\partial p_1}- \frac{\partial \mathsf{H}^{-}_n}{\partial p_{2k-1}}\biggr) \bigl(\zeta (X_{1k}+2\eta)-\zeta (X_{1k}-2\eta)\bigr).
\end{equation*}
\notag
$$
It is clear from (2.12) that this is equivalent to
$$
\begin{equation}
\frac{\partial \mathsf{J}^{-}_n}{\partial X_1}= \sum_{k\ne 1} \biggl(\frac{\partial\mathsf{J}^{-}_n}{\partial p_1}- \frac{\partial \mathsf{J}^{-}_n}{\partial p_{2k-1}}\biggr) \bigl(\zeta (X_{1k}+2\eta)-\zeta (X_{1k}-2\eta)\bigr).
\end{equation}
\tag{4.12}
$$
Repeating the calculation leading to (4.10) for the restriction of $\partial \mathsf{J}_{\pm n}/\partial p_{2k-1}$ to the subspace $\mathcal{P}$ we obtain:
$$
\begin{equation}
\frac{\partial \mathsf{J}_n}{\partial p_{2k-1}}= \frac{\sigma(n\eta)}{\sigma^{n-2m}(\eta)}\sum_{m=0}^{[n/2]}\, \sum_{\substack{\mathcal{I}\cap \mathcal{I}'=\varnothing \\ |\mathcal{I}|=m, \, |\mathcal{I}'|=n-2m}} \Theta (k\in \mathcal{I}) X_{\mathcal{I}'}U_{\mathcal{I}\mathcal{I}'}^{-}
\end{equation}
\tag{4.13}
$$
and
$$
\begin{equation}
\frac{\partial \mathsf{J}_{-n}}{\partial p_{2k-1}}= \frac{\sigma(n\eta)}{\sigma^{n-2m}(\eta)}\sum_{m=0}^{[n/2]}\, \sum_{\substack{\mathcal{I}\cap \mathcal{I}'=\varnothing \\ |\mathcal{I}|=m, \, |\mathcal{I}'|=n-2m}} \Theta (k\in \mathcal{I}\cup \mathcal{I}') X_{\mathcal{I}'} U_{\mathcal{I}\mathcal{I}'}^{+}.
\end{equation}
\tag{4.14}
$$
Here
$$
\begin{equation}
X_{\mathcal{I}'} =\biggl(\,\prod_{j\in \mathcal{I}'}\dot X_j\biggr) \prod_{\substack{j_1, j_2 \in \mathcal{I}'\\ j_1<j_2}}V_{j_1 j_2},
\end{equation}
\tag{4.15}
$$
$$
\begin{equation}
U_{\mathcal{I}\mathcal{I}'}^{\pm} =\prod_{i\in \mathcal{I}}\, \prod_{j\in \mathcal{N}\setminus (\mathcal{I}\cup \mathcal{I}')} U^{\pm}(X_{ij})
\end{equation}
\tag{4.16}
$$
and $\Theta (S)$ is the function equal to $1$ if the statement $S$ is true and to $0$ otherwise. Combining (4.13) and (4.14) we obtain
$$
\begin{equation}
\begin{aligned} \, \nonumber \frac{\partial \mathsf{J}_n^-}{\partial p_{2k-1}}&= \sum_{m=0}^{[n/2]}\kappa_{nm} \biggl\{\,\sum_{\substack{\mathcal{I}\cap \mathcal{I}'=\varnothing \\ |\mathcal{I}|=m, \, |\mathcal{I}'|=n-2m}} \Theta(k\in \mathcal{I}) X_{\mathcal{I}'}(U_{\mathcal{I}\mathcal{I}'}^{-}+ U_{\mathcal{I}\mathcal{I}'}^{+}) \\ &\qquad+\sum_{\substack{\mathcal{I}\cap \mathcal{I}'=\varnothing \\ |\mathcal{I}|=m, \, |\mathcal{I}'|=n-2m}} \Theta (k\in \mathcal{I}') X_{\mathcal{I}'} U_{\mathcal{I}\mathcal{I}'}^{+}\biggr\}, \end{aligned}
\end{equation}
\tag{4.17}
$$
where A similar calculation gives us
$$
\begin{equation}
\frac{\partial \mathsf{J}_{\pm n}}{\partial X_1}= \sum_{m=0}^{[n/2]}\kappa_{nm}\, \sum_{\substack{\mathcal{I}\cap \mathcal{I}'=\varnothing \\ |\mathcal{I}|=m,\, |\mathcal{I}'|=n-2m}} X_{\mathcal{I}'} U_{\mathcal{I}\mathcal{I}'}^{\mp}Z_{\mathcal{I}\mathcal{I}'}^{\pm},
\end{equation}
\tag{4.19}
$$
where
$$
\begin{equation}
\begin{aligned} \, \nonumber Z_{\mathcal{I}\mathcal{I}'}^{+}&=\Theta (1 \in \mathcal{I}) \biggl(\,\sum_{\ell \in \mathcal{N}\setminus(\mathcal{I}\cup\mathcal{I}')} \bigl(\zeta(X_{1\ell}+\eta)-\zeta (X_{1\ell}-\eta)\bigr)\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad+ \sum_{\ell \in \mathcal{I}'}\bigl(\zeta (X_{1\ell}+\eta)- \zeta (X_{1\ell})\bigr)\biggr) \nonumber \\ \nonumber &\qquad+\Theta (1 \in \mathcal{I}\cup \mathcal{I}') \biggl(\,\sum_{\ell \in \mathcal{N}\setminus(\mathcal{I}\cup\mathcal{I}')} \bigl(\zeta (X_{1\ell}+2\eta)-\zeta (X_{1\ell})\bigr)\nonumber\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad+ \sum_{\ell \in \mathcal{I}'}\bigl(\zeta (X_{1\ell}+ 2\eta)- \zeta(X_{1\ell}+\eta)\bigr)\biggr) \nonumber \\ \nonumber &\qquad+\Theta (1 \in \mathcal{N}\setminus \mathcal{I}) \biggl(\,\sum_{\ell \in \mathcal{I}}\,\bigl(\zeta (X_{1\ell}+ 2\eta)- \zeta (X_{1\ell})\bigr)\nonumber\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad+ \sum_{\ell \in \mathcal{I}'} \bigl(\zeta (X_{1\ell}+ 2\eta)-\zeta (X_{1\ell}+\eta)\bigr)\biggr) \nonumber \\ &\qquad+\Theta (1 \in \mathcal{N}\setminus (\mathcal{I}\cup\mathcal{I}')) \biggl(\,\sum_{\ell \in \mathcal{I}}\, \bigl(\zeta(X_{1\ell}+\eta)-\zeta(X_{1\ell}-\eta)\bigr)\nonumber\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad+ \sum_{\ell \in \mathcal{I}'}\bigl(\zeta (X_{1\ell}+ \eta)- \zeta(X_{1\ell})\bigr)\biggr) \end{aligned}
\end{equation}
\tag{4.20}
$$
and $Z_{\mathcal{I}\mathcal{I}'}^{-}$ is obtained from $Z_{\mathcal{I}\mathcal{I}'}^{+}$ by the change $\eta \to -\eta$. This expression can be taken to a more convenient form by using the obvious relations
$$
\begin{equation*}
\Theta(1 \in \mathcal{I}\cup \mathcal{I}')= \Theta(1 \in \mathcal{I})+\Theta(1 \in \mathcal{I}'), \quad \Theta(1 \in \mathcal{N}\setminus \mathcal{I})=1-\Theta(1 \in \mathcal{I}).
\end{equation*}
\notag
$$
The right-hand sides of (4.17) and (4.19) are sums over $m=0,\dots,[n/2]$. We denote the $m$th terms of these sums by $\dfrac{\partial \mathsf{J}_{n,m}^-}{\partial p_{2k-1}}$ and $\dfrac{\partial \mathsf{J}_{\pm n,m}^-}{\partial X_1}$, respectively. We are going to show that
$$
\begin{equation}
\frac{\partial \mathsf{J}_{n,m}}{\partial X_1}- \frac{\partial \mathsf{J}_{-n,m}}{\partial X_1}= \sum_{k\ne 1}\biggl(\frac{\partial \mathsf{J}^{-}_{n,m}}{\partial p_1}- \frac{\partial \mathsf{J}^{-}_{n,m}}{\partial p_{2k-1}}\biggr) \bigl(\zeta(X_{1k}+2\eta)-\zeta (X_{1k}-2\eta)\bigr),
\end{equation}
\tag{4.21}
$$
from which (4.12) follows. A straightforward calculation yields
$$
\begin{equation}
\begin{aligned} \, \nonumber &\kappa_{nm}^{-1}\biggl\{\frac{\partial \mathsf{J}_{n,m}^{-}}{\partial X_1}- \sum_{k\ne 1}\biggl(\frac{\partial\mathsf{J}^{-}_{n,m}}{\partial p_1}- \frac{\partial \mathsf{J}^{-}_{n,m}}{\partial p_{2k-1}}\biggr) \bigl(\zeta (X_{1k}+2\eta)-\zeta(X_{1k}-2\eta)\bigr)\biggr\} \\ \nonumber &\qquad=\sum_{\mathcal{I}\cap \mathcal{I}'= \varnothing} X_{\mathcal{I}'}\biggl\{\Theta (1 \in \mathcal{I}) \biggl[U_{\mathcal{I}\mathcal{I}'}^{-}\sideset{}{'}\sum_\ell \zeta^{-}(X_{1\ell}) \\ \nonumber &\qquad\qquad-U_{\mathcal{I}\mathcal{I}'}^{+} \sideset{}{'}\sum_\ell\zeta^{+}(X_{1\ell})+ U_{\mathcal{I}\mathcal{I}'}^{+}\sideset{}{'}\sum_{\ell\in \mathcal{I}} \zeta^{+}(X_{1\ell}) \\ \nonumber &\qquad\qquad+U_{\mathcal{I}\mathcal{I}'}^{+} \sideset{}{'}\sum_{\ell\in \mathcal{I}} \zeta^{-}(X_{1\ell})-U_{\mathcal{I}\mathcal{I}'}^{-} \sideset{}{'}\sum_{\ell\in \mathcal{I}}\zeta^{-}(X_{1\ell})- U_{\mathcal{I}\mathcal{I}'}^{-} \sideset{}{'}\sum_{\ell\in \mathcal{I}}\zeta^{+}(X_{1\ell}) \\ \nonumber &\qquad\qquad+U_{\mathcal{I}\mathcal{I}'}^{+}\sum_{\ell\in \mathcal{I}'} \zeta^{+}(X_{1\ell})-U_{\mathcal{I}\mathcal{I}'}^{-} \sum_{\ell\in \mathcal{I}'}\zeta^{-}(X_{1\ell})\biggr] \\ &\qquad\qquad+\Theta (1 \in \mathcal{I}') \biggl[U_{\mathcal{I}\mathcal{I}'}^{+}\sum_{\ell\in \mathcal{I}} \zeta^{-}(X_{1\ell})-U_{\mathcal{I}\mathcal{I}'}^{-} \sum_{\ell\in \mathcal{I}}\zeta^{+}(X_{1\ell})\biggr]\nonumber \\ &\qquad\qquad+\sum_{k\ne 1}\Theta (k \in \mathcal{I}) \biggl[ U_{\mathcal{I}\mathcal{I}'}^{-} \zeta^{+}(X_{1k})- U_{\mathcal{I}\mathcal{I}'}^{+} \zeta^{-}(X_{1k})\biggr]\biggr\}, \end{aligned}
\end{equation}
\tag{4.22}
$$
where
$$
\begin{equation}
\zeta^{\pm}(X)=\zeta(X\pm 2\eta)+\zeta(X\mp\eta)-\zeta(X+\pm\eta)-\zeta(X)
\end{equation}
\tag{4.23}
$$
and $\sideset{}{'}\sum_\ell$ means that $\ell \ne 1$. Lemma 4.2. The following identity holds: Proof. This is the $X_1$-derivative of identity (4.11) from Lemma 4.1 for $\mathcal{N}'=\mathcal{N}\setminus \mathcal{I}'$. $\Box$ Using this identity it is easy to see that the right-hand side of (4.22) is zero. Thus, the invariance of the subspace $\mathcal{P}$ of pairs with respect to the flows with Hamiltonians $\mathsf{H}^{-}_n$ is proved. So far, we have considered the restriction of $\mathsf{J}_n$ for $n<N/2=N_0$. The case $N/2 <n\leqslant N$ can be considered in a similar way, with the result that the restriction of $\mathsf{J}_n$ for $N_0 <n\leqslant 2N_0$ is $J_{n-2N_0}$. The proof of Theorem 4.1 can be extended to this case, too. Finally, let us comment on whether or not the integrals of motion are in involution. As the Hamiltonian structure (if any) of the deformed RS system is not known, we are not able to calculate the Poisson brackets between the integrals of motion and prove that they are equal to zero. Our integrals of motion are functions of coordinates and velocities rather than coordinates and momenta. However, in any integrable system all integrals of motion that are in involution are conserved under the flows generated by any one of them. Each higher Hamiltonian $\mathsf{H}^{-}_n$ of the RS system defines a flow $\partial_{T_n^-}$ on the ‘phase space’ $\mathcal{P}$ of the deformed RS system. From the fact that the RS integrals of motion are in involution it follows that the restrictions $H_n$ of the RS Hamiltonians to the space $\mathcal{P}$ are conserved under all $\partial_{T_k^-}$-flows. In this sense we can say that the integrals of motion $H_n$ and $J_n$ of the deformed RS system are in involution. 4.2. The generating function of the integrals of motionIt is known that the integrals of motion of the RS system with $2N$ particles can be unified into a generating function, which is the determinant of the $2N\times 2N$ matrix $zI - L(u)$, where $I$ is the unity matrix, $z$ is the spectral parameter, and $L(u)$ is the Lax matrix depending on another spectral parameter $u$. The Lax matrix has the form where the function $\phi (x, u )$ is given by
$$
\begin{equation}
\phi(x,u)=\frac{\sigma(x+u)}{\sigma(u)\sigma(x)}\,.
\end{equation}
\tag{4.26}
$$
Proposition 4.1 (see [9]). The equality The proof of this proposition is based on the known formula for the determinant of the elliptic Cauchy matrix:
$$
\begin{equation}
\det_{1\leqslant i,j\leqslant n}\phi (y_i-x_j)= \frac{\sigma\bigl(u+\sum_{k=1}^n (y_k-x_k)\bigr)}{\sigma(u)}\, \frac{\prod_{k<l}\sigma(y_k-y_l)\sigma(x_l-x_k)} {\prod_{k,l}\sigma(y_k-x_l)}\,.
\end{equation}
\tag{4.28}
$$
In this subsection we are going to construct the generating function for the integrals of motion (1.7). The idea is to restrict the Lax matrix (4.25) to the subspace $\mathcal{P}$. However, the direct restriction is not possible because some matrix elements become infinite. Nevertheless, we will see that the determinant (4.27) is finite. To regularize the Lax matrix we put and let $\varepsilon \to 0$ at the end. For $\varepsilon =0$ we have $\partial_{t_1}x_{2i}=\dot X_i$ and $\partial_{t_1}x_{2i-1}=0$. To proceed, we need to find $\partial_{t_1}x_{2i-1}$ up to the first non-vanishing order in $\varepsilon$. A simple calculation shows that
$$
\begin{equation}
\partial_{t_1}x_{2i-1}=\varepsilon \sigma(2\eta) \dot X_i^{-1}U_i^- +O(\varepsilon^2),
\end{equation}
\tag{4.30}
$$
where $U_i^-$ is given by (3.13) (with $N_0\to N$). The further calculation of matrix elements of the Lax matrix is straightforward:
$$
\begin{equation}
\begin{aligned} \, L_{2i-1, 2j-1}&:=L_{ij}^{({\rm oo})}= \varepsilon \sigma(2\eta)\dot X_i^{-1}U_i^-\phi (X_{ij}-\eta,u)+ O(\varepsilon^2), \\ L_{2i-1, 2j}&:=L_{ij}^{({\rm oe})}= \varepsilon \sigma(2\eta)\dot X_i^{-1}U_i^-\phi(X_{ij}-2\eta,u)+ O(\varepsilon^2), \\ L_{2i, 2j-1}&:=L_{ij}^{({\rm eo})}= \dot X_i\phi(X_{ij}+\varepsilon,u)+\delta_{ij}O(1)+O(\varepsilon), \\ L_{2i,2j}&:=L_{ij}^{({\rm ee})}=\dot X_i\phi(X_{ij}-\eta,u)+O(\varepsilon). \end{aligned}
\end{equation}
\tag{4.31}
$$
After renumbering the rows and columns, the Lax matrix can be represented as a $ 2\times 2 $ block matrix:
$$
\begin{equation}
L=\begin{pmatrix} L_{ij}^{({\rm oo})} & L_{ij}^{({\rm oe})} \\ L_{ij}^{({\rm eo})} & L_{ij}^{({\rm ee})} \end{pmatrix}, \qquad i,j=1,\dots,N.
\end{equation}
\tag{4.32}
$$
We see that $L_{ii}^{({\rm eo})}$ is singular as $\varepsilon \to 0$ since $\phi(\varepsilon, u)=\varepsilon^{-1}+O(1)$. Using the formula for the determinant of a block matrix we have:
$$
\begin{equation*}
\det(zI-L)=\det \bigl(zI-L^{({\rm oo})}\bigr) \det\bigl(zI-L^{({\rm ee})}-L^{({\rm eo})} (zI-L^{({\rm oo})})^{-1}L^{({\rm oe})}\bigr).
\end{equation*}
\notag
$$
It is easy to see that the right-hand side is finite as $\varepsilon \to 0$. In order to find the limit as $\varepsilon \to 0$ we can put $L_{ij}^{({\rm oo})}=0$ and forget about the next-to-leading powers of $\varepsilon$ in other blocks. In this way we find that
$$
\begin{equation*}
\lim_{\varepsilon \to 0}\bigl(L^{({\rm eo})}(zI-L^{({\rm oo})})^{-1} L^{({\rm oe})}\bigr)_{ij}=\sigma(2\eta)z^{-1}U_i^-\phi (X_{ij}-2\eta,u).
\end{equation*}
\notag
$$
Therefore, the generating function of integrals of motion is
$$
\begin{equation}
R(z,u )=\det_{1\leqslant i,j\leqslant N} \bigl(z\delta_{ij}- \dot X_i \phi(X_{ij}-\eta,u )- \sigma(2\eta)z^{-1} U_i^- \phi(X_{ij}-2\eta,u )\bigr).
\end{equation}
\tag{4.33}
$$
Proposition 4.2. The generating function $R(z, u)$ is given by Sketch of proof. The proof is a lengthy but straightforward calculation, which uses the formula for the determinant of the sum of two matrices and the formula for the determinant of the elliptic Cauchy matrix (4.28). Here are some details. First of all, the determinant $\det (I+M)$ is equal to the sum of all diagonal minors of all sizes of the matrix $M$, including the ‘empty minor’ which is put equal to $1$. After that we encounter determinants of the form $\det (A_\mathcal{J} +B_{\mathcal{J}})$, where $A_\mathcal{J}$ and $B_{\mathcal{J}}$ are the diagonal minors of size $n\leqslant N$ of the matrices $\dot X_i \phi(X_{ij}-\eta, u)$ and $\sigma(2\eta)z^{-1} U_i^- \phi(X_{ij}- 2\eta, u)$, respectively, whose rows and columns have indices from the set The characteristic equation defines a Riemann surface $\widetilde \Gamma$ which is a $2N$-sheeted covering of the $u$-plane. This Riemann surface is an integral of motion. A point in it is a pair $P=(z, u )$, where $z$ and $ u$ are connected by (4.35). There are $2N$ points over each point $u$. It is easy to see from the right-hand side of (4.34) that $\widetilde \Gamma$ is invariant under the simultaneous transformations
$$
\begin{equation}
u \mapsto u +2\omega, \quad z\mapsto e^{-2\zeta (\omega)\eta}z \quad \text{and} \quad u \mapsto u +2\omega', \quad z\mapsto e^{-2\zeta (\omega')\eta}z.
\end{equation}
\tag{4.36}
$$
The quotient of $\widetilde \Gamma$ by the transformations (4.36) is an algebraic curve $\Gamma$, which covers an elliptic curve with periods $2\omega$ and $2\omega'$. It is the spectral curve of the deformed RS model. Proposition 4.3. The spectral curve $\Gamma$ admits a holomorphic involution $\iota$ with two fixed points. Proof. It was proved in the previous subsection that $J_{-k}=J_k$. Therefore, the equation $R(z, u )=0$ is invariant under the involution as is easily seen from (4.34). The fixed points lie over the points $u_{*}$ such that $u_{*}=2N\eta-u_{*}$ modulo the lattice with periods $2\omega$ and $2\omega '$, that is, $u_{*}=N\eta-\omega_{\alpha}$, where $\omega_{\alpha}$ is either $0$ or one of the three half-periods Substituting this into the equation of the spectral curve and taking into account that $J_{-k}=J_k$, we conclude that the fixed points are $(\pm1,N\eta)$ and there are no fixed points over $u_{*}=N\eta-\omega_{\alpha}$ for $\omega_{\alpha}\ne 0$. $\Box$
5. The Bäcklund transformationSince [10]–[14] it became a common knowledge among experts that integrable many- body systems of CM and RS type are dynamical systems for poles of singular solutions of nonlinear integrable differential and difference equations. The nonlinear integrable equations are known to serve as the compatibility conditions for linear differential or difference equations for the ‘wave function’ $\psi$. Poles of solutions of nonlinear equations (zeros of the tau-function) are simultaneously poles of the $\psi$-function, so the latter obey equations of motion of CM or RS type. In fact zeros of the $\psi$-function obey the same equations, and this gives rise to the idea to obtain the Bäcklund transformation of the CM or RS system as a transition from poles to zeros. This method works well for all previously known examples, and we are going to apply it to the case of the Toda lattice with constraint of type B. The first linear problem for the Toda lattice with constraint of type B has the form [17]
$$
\begin{equation}
\partial_t \psi(x)=v(x)\bigl(\psi(x+\eta)-\psi(x-\eta)\bigr),
\end{equation}
\tag{5.1}
$$
where $v(x)$ is expressed in terms of the tau-function $\tau (x)$ as For elliptic solutions the tau-function has the form where all zeros $x_j$ are assumed to be distinct, so that $v(x)$ is an elliptic function with periods $2\omega$ and $2\omega'$. Therefore, solutions to (5.1) can be sought as double-Bloch functions, that is, functions such that
$$
\begin{equation*}
\psi(x+2\omega)=B\psi(x) \quad\text{and}\quad \psi(x+2\omega')=B'\psi(x)
\end{equation*}
\notag
$$
for some Bloch multipliers $B$ and $B'$. Poles of the $\psi$-function are zeros of the tau-function, so we can represent solutions of (5.1) in the form
$$
\begin{equation}
\psi(x)=\mu^{x/\eta}\exp\bigl((\mu-\mu^{-1})t\bigr)\, \frac{\hat\tau(x)}{\tau(x)}\,,
\end{equation}
\tag{5.4}
$$
where for some $y_i$. Then the $\psi$-function is indeed a double-Bloch function with Bloch multipliers
$$
\begin{equation*}
B=\mu^{2\omega/\eta}\exp\biggl(2\zeta(\omega)\sum_{j=1}^N(x_j-y_j)\biggr)
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
B'=\mu^{2\omega'/\eta}\exp\biggl(2\zeta(\omega') \sum_{j=1}^N(x_j-y_j)\biggr).
\end{equation*}
\notag
$$
Below we will see (in Proposition 5.1) that so that the Bloch multipliers do not depend on time. It was proved in [16] that poles of the $\psi$-function $x_j$ satisfy the equations of motion (1.3) for any $\mu$. Theorem 5.1. The zeros $y_i$ and the poles $x_i$ of the $\psi$-function satisfying equation (5.1) obey the system of equations These are so-called self-dual equations of motion. They are symmetric under the change
$$
\begin{equation*}
x_j \leftrightarrow y_j,\quad \mu \leftrightarrow \mu^{-1}.
\end{equation*}
\notag
$$
The transition $x_j \to y_j$ can be regarded as a Bäcklund transformation of the deformed RS system. Proof of Theorem 5.1. Substituting ansatz (5.4) into equation (5.1) for $v(x)$ given by (5.2) we obtain the equation
$$
\begin{equation}
\frac{\partial_t\hat\tau(x)}{\hat\tau(x)}-\frac{\partial_t\tau(x)}{\tau(x)}+ \mu-\mu^{-1}=\mu\frac{\hat\tau(x+\eta)\tau(x-\eta)}{\hat\tau(x)\,\tau(x)}- \mu^{-1}\frac{\tau(x+\eta)\hat\tau(x-\eta)}{\tau(x)\, \hat\tau(x)}\,.
\end{equation}
\tag{5.7}
$$
It is obvious from this equation that it is invariant under the simultaneous changes
$$
\begin{equation*}
\tau \leftrightarrow \hat\tau,\quad \mu \leftrightarrow \mu^{-1},
\end{equation*}
\notag
$$
so the $y_j$ satisfy the same equations of motion (1.3). Both sides of (5.7) have simple poles at $x=x_j$ and $x=y_j$. Equating the residues we obtain (5.6). $\Box$ We note that the equations of motion (1.4) for the $x_j$ and $y_j$ can in principle can be derived from (5.6). To do this one must take the time derivative of these equations and use them again by substituting the expressions for $\dot x_j$ and $\dot y_j$ in terms of $x_j$ and $y_j$. Equations (1.4) turn out to be equivalent to a non-trivial identity for multi-variable elliptic functions, which is too complicated to be proved directly. However, we do not need to prove it directly since the equations of motion for the $x_j$ follow from the result of [16] and the ones for the $y_j$ follow from the symmetry $x_j \leftrightarrow y_j$. Note that the Bäcklund transformation for the RS system differs from (5.6) by the absence of second terms on the right-hand sides. In this sense it is contained in (5.6) as a formal limiting case as $\mu \to \infty$ (or $\mu \to 0$). This follows from (5.6). The proof is based on higher-degree identities of Fay type for the $\sigma$-function (see [24], for instance). It is almost the same as the corresponding proof from [24], so we leave it out here. 6. The discrete-time dynamics and continuum limitsThe Bäcklund transformation $x_j \to y_j$ can be regarded as one-step time evolution in discrete time. Denoting the discrete time variable by $n\in \mathbb{Z}$, we put $x_i =x_i^n$ and $y_i=x_i^{n+1}$. Shifting $n\to n-1$ in the second equation in (5.6), so that the left-hand sides of the two equations become equal, we conclude that the right-hand sides are equal too, which results in equations (1.12), or
$$
\begin{equation}
\begin{aligned} \, \nonumber &\prod_{j=1}^N\frac{\sigma(x_i^{n}-x_j^{n+1}) \sigma(x_i^{n}-x_j^{n}+\eta)\sigma(x_i^{n}-x_j^{n-1}-\eta)} {\sigma(x_i^{n}-x_j^{n+1}+\eta)\sigma(x_i^{n}-x_j^{n}-\eta) \sigma(x_i^{n}-x_j^{n-1})} \\ \nonumber &\qquad=-1+\mu^{-2}\prod_{j=1}^N \frac{\sigma(x_i^{n}-x_j^{n+1})\sigma(x_i^{n}- x_j^{n-1}+\eta)} {\sigma(x_i^{n} -x_j^{n+1}+\eta)\sigma(x_i^{n}-x_j^{n-1})} \\ &\qquad\qquad\;\;+ \mu^{-2}\prod_{j=1}^N \frac{\sigma(x_i^{n}-x_j^{n+1}-\eta)\sigma(x_i^{n}-x_j^{n}+\eta)} {\sigma(x_i^{n} -x_j^{n+1}+\eta)\sigma(x_i^{n}-x_j^{n}-\eta)}\,. \end{aligned}
\end{equation}
\tag{6.1}
$$
Let us discuss the continuous-time limit of equations (6.1). In fact they admit different continuum limits. For one of them we introduce the variables and assume that these variables vary smoothly with time, that is,
$$
\begin{equation*}
X_j^{n+1}=X_j^n +O(\varepsilon)\quad \text{as } \varepsilon \to 0,
\end{equation*}
\notag
$$
where we introduce the lattice spacing $\varepsilon$ in the time lattice, so that the continuous- time variable is $t=n\varepsilon$. In terms of the variables $X_j^n$ equations (6.1) acquire the form
$$
\begin{equation}
\begin{aligned} \, \nonumber &\prod_{j=1}^N\frac{\sigma(X_i^{n}-X_j^{n+1}-\eta) \sigma(X_i^{n}-X_j^{n}+\eta)\sigma(X_i^{n}-X_j^{n-1})} {\sigma(X_i^{n}-X_j^{n+1})\sigma(X_i^{n}-X_j^{n}-\eta) \sigma(X_i^{n}-X_j^{n-1}+\eta)} \\ \nonumber &\qquad=-1+\mu^{-2}\prod_{j=1}^N \frac{\sigma(X_i^{n}-X_j^{n+1}-\eta)\sigma(X_i^{n}-X_j^{n-1}+2\eta)} {\sigma(X_i^{n} -X_j^{n+1})\sigma(X_i^{n}- X_j^{n-1}+\eta)}\notag \\ &\qquad\qquad+\mu^{-2}\prod_{j=1}^N \frac{\sigma(X_i^{n}-X_j^{n+1}-2\eta)\sigma(X_i^{n}-X_j^{n}+\eta)} {\sigma(X_i^{n} -X_j^{n+1}) \sigma(X_i^{n}-X_j^{n}-\eta)}\,. \end{aligned}
\end{equation}
\tag{6.3}
$$
We must expand these equations in powers of $\varepsilon$ while taking into account that
$$
\begin{equation*}
X_j^{n\pm 1}=X_j \pm \varepsilon \dot X_j+ \frac{1}{2}\,\varepsilon^2 \ddot X_j+O(\varepsilon^3)
\end{equation*}
\notag
$$
as $\varepsilon \to 0$. The expansion procedure is straightforward. It is enough to expand up to order $\varepsilon$. For the consistency of the expansion procedure one should assume that $\mu^{-1}$ is of order $\varepsilon$. Putting $\mu^{-1}=\varepsilon$ one obtains equations (1.4) for the $X_j$ in the leading order of $\varepsilon$. Another possibility is to assume that the original variables $x_j^n$ are smooth with respect to time, that is,
$$
\begin{equation*}
x_j^{n\pm 1}=x_j \pm \varepsilon \dot x_j+ \frac{1}{2}\, \varepsilon^2 \ddot x_j +O(\varepsilon^3).
\end{equation*}
\notag
$$
In this case one should expand equations (6.1). It is easy to see that in general position, that is, when $\mu^{-2}-1 =O(1)$ as $\varepsilon \to 0$, the leading order is $\varepsilon$ and the expansion gives rise to the RS equations (1.2). However, if $\mu^{-2} -1 =O(\varepsilon )$, say, $\mu^{-2} =1+\alpha \varepsilon +O(\varepsilon^2)$, then the first order gives us the identity $0=0$ and one should expand up to the second order in $\varepsilon$. This procedure is rather cumbersome but straightforward. As a result, one obtains the equations derived in [18] for the dynamics of poles of elliptic solutions to the semi-discrete BKP equation:
$$
\begin{equation}
\begin{aligned} \, \nonumber &\sum_{j\ne i}(\ddot x_i \dot x_j -\dot x_i\ddot x_j) \biggl(\frac{\wp '(\eta)}{\wp (x_{ij})-\wp (\eta)}-2\zeta(\eta)\biggr) \\ \nonumber &\qquad+\sum_k\,\sum_{j\ne i,k}\dot x_i \dot x_j \dot x_k \, \frac{\wp '(x_{ij})}{\wp (x_{ij})-\wp (\eta)} \biggl(\frac{\wp'(\eta)}{\wp (x_{ij})-\wp(\eta)}-2\zeta(\eta)\biggr) \\ \nonumber &\qquad-2\zeta(\eta)\sum_{j\ne i} \dot x_i \dot x_j^2\, \frac{\wp '(x_{ij})}{\wp (x_{ij})-\wp (\eta)} \\ &\qquad-\alpha\biggl(\ddot x_i+\sum_{j\ne i}\dot x_i \dot x_j \bigl(\zeta(x_{ij}+\eta)+\zeta(x_{ij}-\eta)-2\zeta (x_{ij})\bigr)\biggr)=0 \end{aligned}
\end{equation}
\tag{6.4}
$$
(here we correct the mistake in [18] — the sum in the third line was missing there). In contrast to the equations of motion discussed previously, these equations are not resolved with respect to the $\ddot x_j$s.
7. The discrete-time deformed RS system from fully discrete BKP equationIn this section we show that the discrete-time elliptic deformed RS system (1.12) is a dynamical system for poles of elliptic solutions to the fully discrete BKP equation [29]. We begin with the continuous BKP hierarchy [30], [31]. Let $\mathbf{t}=\{t_1,t_3,t_5,\dots\}$ be an infinite set of continuous ‘times’ indexed by odd integers. They are independent variables of the hierarchy. The dependent variable is the tau-function $\tau=\tau(\mathbf{t})$. The BKP hierarchy is encoded in the generating basic bilinear relation for the tau-function [31]:
$$
\begin{equation}
\oint_{C_{\infty}}\frac{dz}{2\pi i z}\, \exp\bigl(\xi (\mathbf{t},z)- \xi(\mathbf{t}', z)\bigr) \tau (\mathbf{t}-2[z^{-1}])\, \tau (\mathbf{t}' +2[z^{-1}])=\tau(\mathbf{t})\tau(\mathbf{t}'),
\end{equation}
\tag{7.1}
$$
which is valid for all $\mathbf{t}$ and $\mathbf{t}'$. Here we use the standard notation
$$
\begin{equation*}
\xi(\mathbf{t},z)=\sum_{k=1,3,5,\dots}t_k z^k \quad\text{and}\quad \mathbf{t}\pm 2[z^{-1}]=\biggl\{t_1 \pm \frac{2}{z}\,,t_3 \pm \frac{2}{z^3}\,,t_5 \pm \frac{2}{z^5}\,,\dots\biggr\}.
\end{equation*}
\notag
$$
The contour $C_{\infty}$ is a large circle around the point at infinity such that the singularities coming from factors containing tau-functions are inside it and those coming from the exponential factor are outside. The discrete BKP equation is obtained as follows. Put
$$
\begin{equation}
\tau({\ell },m,n):=\tau(\mathbf{t}-2{\ell}[a^{-1}]-2m[b^{-1}]-2n[c^{-1}])
\end{equation}
\tag{7.2}
$$
and consider this function as a function of the discrete variables $\ell$, $m$, and $n$. Proposition 7.1 (see [29]). The function $\tau(\ell,m,n)$ satisfies the equation This is the fully discrete BKP equation, which appeared originally in Miwa’s paper [29]. Proof of Proposition 7.1. Setting
$$
\begin{equation*}
t_k'=t_k-\frac{2a^{-k}}{k}-\frac{2b^{-k}}{k}- \frac{2c^{-k}}{k}
\end{equation*}
\notag
$$
in the bilinear relation, one can calculate the integral on the left-hand side of (7.1) with the help of residue calculus. (In doing so one must take into account that the poles at the points $a$, $b$, and $c$ which arise from the exponential factor lie outside the contour, and shrink the contour to infinity.) As a result, one obtains equation (7.3). $\Box$ Let us introduce the wave function $\psi(\ell,m;z)$ by the formula
$$
\begin{equation}
\psi({\ell },m;z)=e^{\xi(\mathbf{t},z)}\biggl(\frac{a-z}{a+z}\biggr)^{\ell} \biggl(\frac{b-z}{b+z}\biggr)^{m}\,\frac{\tau(\mathbf{t}-2{\ell}[a^{-1}]- 2m[b^{-1}]-2[z^{-1}])}{\tau(\mathbf{t}-2l[a^{-1}]-2m[b^{-1}])}\,.
\end{equation}
\tag{7.4}
$$
Proposition 7.2. The function $\psi(\ell,m;z)$ obeys the linear equation where Proof. Substituting (7.4) into (7.5), one sees that (7.5) is equivalent to the discrete BKP equation (7.3). $\Box$ So far, all discrete variables have entered symmetrically. Now we are going to break this symmetry by distinguishing one of the variables, for instance, $\ell$, and treating it as the space variable on the space lattice with lattice spacing $\eta$. Accordingly, we introduce the continuous variable $x={\ell}\eta$ with $\ell \pm 1$ corresponding to $x\pm \eta$. The discrete time variable is $m$. It is convenient to change the notation as follows:
$$
\begin{equation*}
\tau(\ell,m,n)\to \tau^{m}(x)\quad\text{and} \quad \psi(\ell,m)\to\psi^{m}(x)
\end{equation*}
\notag
$$
(the variable $n$ is supposed to be fixed). With this notation, equation (7.5) assumes the form
$$
\begin{equation}
\psi^{m+1}(x)-\psi^m(x+\eta)= \kappa u^m(x)\bigl(\psi^{m+1}(x+\eta)-\psi^m(x)\bigr),
\end{equation}
\tag{7.7}
$$
$$
\begin{equation}
u^m(x)=\frac{\tau^m(x)\tau^{m+1}(x+\eta)} {\tau^{m}(x+\eta)\tau^{m+1}(x)}\,, \quad \kappa=\frac{a-b}{a+b}\,.
\end{equation}
\tag{7.8}
$$
Let us point out how the two continuous time limits discussed in the previous section look like in these terms. For the first limit (which leads to the Toda lattice with constraint of type B) one lets $b\to a$ and $m\to \infty$ in such a way that the continuous time variable $t=m(b^{-1}-a^{-1})$ remains finite and non-zero. For the second limit (which leads to the semi-discrete BKP equation) one lets $b\to \infty$ and $m\to \infty$ in such a way that the continuous time variable $t=mb^{-1}$ remains finite and non-zero. Now we are ready to obtain the equations of motion for poles in $x$ of the wave functions $\psi^m(x)$ (which are zeros of the $\tau^m(x)$) as functions of the discrete time $m$. Theorem 7.1. For solutions elliptic in $x$, poles of the wave function $\psi^m(x)$ (zeros of $\tau^m(x)$) obey the equations of motion of the deformed RS model in discrete time (1.12). Proof. In accordance with (7.4), we represent the wave function in the form
$$
\begin{equation}
\psi^m(x)=\lambda^{x/\eta}\nu^m\,\frac{\hat\tau^m(x)}{\tau^m(x)}\,,
\end{equation}
\tag{7.9}
$$
where
$$
\begin{equation*}
\lambda=\frac{a-z}{a+z}\quad\text{and}\quad \nu=\frac{b-z}{b+z}\,.
\end{equation*}
\notag
$$
Note that in terms of $\lambda$ and $\nu$ we have For elliptic solutions we set
$$
\begin{equation}
\tau^m(x)=\prod_{j=1}^N \sigma(x-x_j^m)\quad\text{and} \quad \hat \tau^m(x)=\prod_{j=1}^N \sigma(x-y_j^m)
\end{equation}
\tag{7.11}
$$
and assume that all the $x_j^m$ are distinct. Plugging (7.9) into the linear equation (7.7) we obtain the following equation connecting $\tau^m(x)$ and $\hat \tau^m(x)$:
$$
\begin{equation}
\nu\,\frac{\hat\tau^{m+1}(x)}{\tau^{m+1}(x)}- \lambda\,\frac{\hat \tau^{m}(x+\eta)}{\tau^{m}(x+\eta)}= \lambda \nu \kappa\frac{\tau^m(x) \hat\tau^{m+1}(x+\eta)} {\tau^{m}(x+\eta)\tau^{m+1}(x)}- \kappa\frac{\hat \tau^m(x) \tau^{m+1}(x+\eta)} {\tau^{m}(x+\eta)\tau^{m+1}(x)}\,.
\end{equation}
\tag{7.12}
$$
Both sides have simple poles at $x=x_i^{m+1}$ and $x=x_i^m-\eta$. Equating the residues at these poles and making the shift $m\to m-1$ when necessary we obtain the equations
$$
\begin{equation}
\nu \tau^{m-1}(x_i^m+\eta)\hat\tau^m(x_i^m)= \lambda \nu \kappa \tau^{m-1}(x_i^m)\hat\tau^m(x_i^m+\eta)- \kappa \tau^m(x_i^m+\eta)\hat\tau^{m-1}(x_i^m)
\end{equation}
\tag{7.13}
$$
and
$$
\begin{equation}
\lambda\tau^{m+1}(x_i^m -\eta)\hat \tau^m(x_i^m)= \kappa\tau^{m+1}(x_i^m)\hat \tau^m(x_i^m-\eta)- \lambda \nu\kappa \tau^m(x_i^m-\eta)\hat \tau^{m+1}(x_i^m).
\end{equation}
\tag{7.14}
$$
Substituting $x=x_i^m$ and $x=x_i^{m+1}-\eta$ into (7.12), we see that one term vanishes. Making the shift $m\to m-1$ when necessary, we obtain the equations
$$
\begin{equation}
\kappa\tau^{m+1}(x_i^m +\eta)\hat \tau^m(x_i^m)= \lambda\tau^{m+1}(x_i^m)\hat \tau^m(x_i^m+\eta)- \nu\tau^m(x_i^m+\eta)\hat \tau^{m+1}(x_i^m)
\end{equation}
\tag{7.15}
$$
and
$$
\begin{equation}
\lambda \nu \kappa\tau^{m-1}(x_i^m-\eta)\hat\tau^m(x_i^m)= \nu \tau^{m-1}(x_i^m)\hat\tau^m(x_i^m-\eta)- \lambda \tau^m(x_i^m-\eta)\hat\tau^{m-1}(x_i^m).
\end{equation}
\tag{7.16}
$$
Equations (7.13) and (7.15) can be regarded as a system of linear equations for $\hat \tau^m(x_i^m)$ and $\hat\tau^m(x_i^m+\eta)$. According to Cramer’s rule, the solution $\hat\tau^m(x_i^m)$ is as follows:
$$
\begin{equation}
\hat \tau^m(x_i^m)=-\kappa \tau^m(x_i^m+\eta)\frac{\begin{vmatrix} \hat \tau^{m-1}(x_i^m) & \lambda \nu \tau^{m-1}(x_i^m) \\ \nu \hat \tau^{m+1}(x_i^m) & \lambda \tau^{m+1}(x_i^m) \end{vmatrix}}{\begin{vmatrix} \nu \tau^{m-1}(x_i^m+\eta) & -\lambda \nu \kappa\tau^{m-1}(x_i^m) \\ -\kappa \tau^{m+1}(x_i^m+\eta) & \lambda \tau^{m+1}(x_i^m) \end{vmatrix}}\,.
\end{equation}
\tag{7.17}
$$
In their turn, equations (7.14) and (7.16) can be regarded as a system of linear equations for $\hat \tau^m(x_i^m)$ and $\hat\tau^m(x_i^m-\eta)$. The solution for $\hat\tau^m(x_i^m)$ is
$$
\begin{equation}
\hat\tau^m(x_i^m)=-\kappa \tau^m(x_i^m-\eta)\frac{\begin{vmatrix} \lambda \nu \hat \tau^{m+1}(x_i^m) & \tau^{m+1}(x_i^m) \\ \lambda \hat \tau^{m-1}(x_i^m) & \nu \tau^{m-1}(x_i^m) \end{vmatrix}}{\begin{vmatrix} -\lambda \tau^{m+1}(x_i^m-\eta) & \kappa \tau^{m+1}(x_i^m) \\ \lambda \nu \kappa \tau^{m-1}(x_i^m-\eta) & -\nu\tau^{m-1}(x_i^m) \end{vmatrix}}\,.
\end{equation}
\tag{7.18}
$$
Equating the right-hand sides of (7.17) and (7.18) we obtain the equations of the discrete-time dynamics of the poles $x_i^m$:
$$
\begin{equation}
\begin{aligned} \, \nonumber &\tau^{m+1}(x_i^m)\tau^m(x_i^m-\eta)\tau^{m-1}(x_i^m+\eta)+ \tau^{m+1}(x_i^m-\eta)\tau^m(x_i^m +\eta)\tau^{m-1}(x_i^m) \\ \nonumber &\qquad=\kappa^2\tau^{m+1}(x_i^m+\eta)\tau^m(x_i^m-\eta)\tau^{m-1}(x_i^m) \\ &\qquad\qquad+\kappa^2\tau^{m+1}(x_i^m)\tau^m(x_i^m +\eta) \tau^{m-1}(x_i^m -\eta). \end{aligned}
\end{equation}
\tag{7.19}
$$
These are equations (1.12). $\Box$ 8. A field analogue of the deformed RS system on a space-time latticeIt is known that integrable models of CM and RS type admit extensions to field theories (‘field analogues’) in which the coordinates of particles $x_i$ become ‘fields’ $x_i(x,t)$ depending not only on time $t$ but also on the space variable $x$. The equations of motion of these more general models can be obtained as equations for poles of more general elliptic solutions (called elliptic families in [27]) of nonlinear integrable equations. In this case one considers solutions which are elliptic functions of a linear combination $\lambda$ of higher times $t_k$ of the hierarchy, their poles $\lambda_i(x,t)$ being functions of the space and time variables $x$ and $t$ (in the CM/KP case $x=t_1$ and $t=t_2$). They obey a system of partial differential or difference equations which are the equations of motion of the field analogue of the CM or RS system. They were obtained using this method in [27] and [28], respectively (also see [32], where elliptic families of solutions to the constrained Toda lattice were discussed). In this section we apply this method to the fully discrete BKP equation and obtain a field extension of the deformed RS model on a space-time lattice. Namely, we consider elliptic families of solutions to the fully discrete BKP equation and find dynamical equations for their poles. 8.1. Equations of motion on the space-time lattice derived from elliptic solutions of the fully discrete BKP equationWe begin with the BKP hierarchy (7.1). Let $\lambda=\sum_{j \text{ odd}} \beta_j t_j$ be an arbitrary linear combination of the times of the hierarchy. According to [27], the tau-function $\tau(\lambda,\mathbf{t})$ corresponding to a solution that is an elliptic function of $\lambda$ has the general form
$$
\begin{equation}
\tau(\lambda,\mathbf{t})=\rho (\mathbf{t})\exp(c_1\lambda+c_2\lambda^2) \prod_{j=1}^N \sigma(\lambda-\lambda_j(\mathbf{t})),
\end{equation}
\tag{8.1}
$$
where $\rho(\mathbf{t})$ does not depend on $\lambda$, and $c_1$ and $c_2$ are some constants. Note that the shift of $\lambda$ by any period should give an equivalent tau-function, that is, the tau-function which differs from the initial one by multiplication by the exponential function of a linear form in the times. Therefore, the zeros $\lambda_i$ of the tau-function (poles of the solution) should satisfy the condition
$$
\begin{equation}
\sum_i\lambda_i(\mathbf{t})=\text{ a linear form in $\mathbf{t}$}.
\end{equation}
\tag{8.2}
$$
This means that the ‘centre of mass’ of the set of the $\lambda_i$ moves uniformly with respect to all times. For the fully discrete BKP equation we can consider elliptic solutions of the form
$$
\begin{equation}
\tau^m (\lambda,x)=\rho^m(x)\exp(c_1\lambda+c_2\lambda^2) \prod_{j=1}^N \sigma(\lambda-\lambda_j^m (x)),
\end{equation}
\tag{8.3}
$$
where $x$ is the space variable and $m$ is the discrete time variable. We assume that all the $\lambda_j$ are distinct. Theorem 8.1. The zeros $\lambda_j^m (x)$ of the tau-function $\tau^m(\lambda,x)$ satisfy the system of equations Proof. Set
$$
\begin{equation}
\sigma^m(\lambda,x):=\prod_{j=1}^N\sigma(\lambda-\lambda_j^m(x)).
\end{equation}
\tag{8.6}
$$
We can find solutions of (7.7) in the form
$$
\begin{equation}
\psi^m(x)=\frac{\hat\tau^m(\lambda,x)}{\sigma^m(\lambda,x)}\,.
\end{equation}
\tag{8.7}
$$
Plugging (8.3) and (8.7) into (7.7) we obtain the equation
$$
\begin{equation}
\begin{aligned} \, \nonumber &\frac{\hat\tau^{m+1}(\lambda,x)}{\sigma^{m+1}(\lambda,x)}- \frac{\hat \tau^{m}(\lambda,x+\eta)}{\sigma^{m}(\lambda,x+\eta)} \\ &\qquad=\kappa^m(x)\, \frac{\sigma^{m}(\lambda,x)\hat\tau^{m+1}(\lambda,x+\eta)} {\sigma^{m}(\lambda,x+\eta) \sigma^{m+1}(\lambda,x)}- \kappa^m(x)\,\frac{\sigma^{m+1}(\lambda,x+\eta)\hat \tau^{m}(\lambda,x)} {\sigma^{m}(\lambda, x+\eta) \sigma^{m+1}(\lambda,x)}\,. \end{aligned}
\end{equation}
\tag{8.8}
$$
The further calculation is similar to the one performed in the previous section. Both sides of (8.8) are elliptic functions of $\lambda$ with simple poles at $\lambda =\lambda_i^{m+1}(x)$ and $\lambda =\lambda_i^m(x-\eta )$. Equating the residues at these poles and making the shifts $m\to m-1$ and $x\to x-\eta$ when necessary we obtain the equations
$$
\begin{equation}
\begin{aligned} \, \nonumber &\sigma^{m-1}(\lambda_i^{m}(x),x+\eta)\hat\tau^{m}(\lambda_i^m(x),x)= \kappa^{m-1}(x)\sigma^{m-1}(\lambda_i^{m}(x),x) \hat\tau^{m}(\lambda_i^m(x),x+\eta) \\ &\qquad-\kappa^{m-1}(x)\sigma^{m}(\lambda_i^{m}(x),x+\eta) \hat\tau^{m-1}(\lambda_i^m(x),x) \end{aligned}
\end{equation}
\tag{8.9}
$$
and
$$
\begin{equation}
\begin{aligned} \, \nonumber &\sigma^{m+1}(\lambda_i^{m}(x),x-\eta)\hat\tau^{m}(\lambda_i^m(x),x)= \kappa^{m}(x-\eta)\sigma^{m+1}(\lambda_i^{m}(x),x) \hat\tau^{m}(\lambda_i^m(x),x-\eta) \\ &\qquad-\kappa^{m}(x-\eta)\sigma^{m}(\lambda_i^{m}(x),x-\eta) \hat\tau^{m+1}(\lambda_i^m(x),x). \end{aligned}
\end{equation}
\tag{8.10}
$$
Substituting $\lambda=\lambda_i^m(x)$ and $\lambda=\lambda_i^{m+1}(x-\eta)$ into (8.8) we see that one term vanishes. Making the shifts $m\to m-1$ and $x\to x-\eta$ when necessary, we obtain the equations
$$
\begin{equation}
\nonumber \kappa^{m}(x)\sigma^{m+1}(\lambda_i^{m}(x),x+\eta) \hat\tau^{m}(\lambda^m_i(x),x)=\sigma^{m+1}(\lambda_i^{m}(x),x) \hat\tau^{m}(\lambda_i^m(x),x+\eta)
\end{equation}
\notag
$$
$$
\begin{equation}
\qquad-\sigma^{m}(\lambda_i^{m}(x), x+\eta)\hat\tau^{m+1}(\lambda_i^m(x),x),
\end{equation}
\tag{8.11}
$$
$$
\begin{equation}
\nonumber \kappa^{m-1}(x-\eta)\sigma^{m-1}(\lambda_i^{m}(x),x-\eta) \hat\tau^{m}(\lambda^m_i(x),x)=\sigma^{m-1}(\lambda_i^{m}(x),x) \hat\tau^{m}(\lambda_i^m(x),x-\eta)
\end{equation}
\notag
$$
$$
\begin{equation}
\qquad-\sigma^{m}(\lambda_i^{m}(x),x-\eta)\hat\tau^{m-1}(\lambda_i^m(x),x).
\end{equation}
\tag{8.12}
$$
Consider equations (8.9) and (8.11). They can be regarded as a system of linear equations for $\hat\tau^m(\lambda_i^m(x),x)$ and $\hat\tau^m(\lambda_i^m(x),x+\eta)$. In their turn, (8.10) and (8.12) can be regarded as a system of linear equations for $\hat\tau^m(\lambda_i^m(x),x)$ and $\hat\tau^m(\lambda_i^m(x),x- \eta)$. Solving them for $\hat\tau^m(\lambda_i^m(x),x)$ with the help of Cramer’s rule as in the previous section, and equating the results we obtain equations (8.4). $\Box$ Equations (8.4) are the equations of motion for the field analogue of the deformed RS model on the space-time lattice. They resemble equations (1.12) and reduce to them if we set
$$
\begin{equation*}
\lambda_i^m(x)=x_i^m +x\quad\text{and}\quad \kappa^m(x)=\kappa={\rm const}.
\end{equation*}
\notag
$$
In the limit as $\kappa \to 0$, when the right-hand side vanishes, equations (8.4) reduce to the fully discrete version of the field extension of the RS model [28] (also see [33], where similar equations were obtained from the elliptic Lax pair of general form). Note also that equations (8.4) can fully be written in terms of the function $\tau^m(\lambda,x)$ (8.3):
$$
\begin{equation}
\begin{aligned} \, \nonumber &\tau^{m+1}(\lambda_i^m(x),x)\tau^{m}(\lambda_i^m(x),x-\eta) \tau^{m-1}(\lambda_i^m(x),x+\eta) \\ \nonumber &\qquad\qquad+\tau^{m+1}(\lambda_i^m(x),x-\eta) \tau^{m}(\lambda_i^m(x),x+\eta)\tau^{m-1}(\lambda_i^m(x),x) \\ \nonumber &\qquad=\tau^{m+1}(\lambda_i^m(x),x)\tau^{m}(\lambda_i^m(x),x+\eta) \tau^{m-1}(\lambda_i^m(x), x-\eta) \\ &\qquad\qquad+\tau^{m+1}(\lambda_i^m(x),x+\eta) \tau^{m}(\lambda_i^m(x),x-\eta)\tau^{m-1}(\lambda_i^m(x),x). \end{aligned}
\end{equation}
\tag{8.13}
$$
In the next subsection we discuss continuous-time limits of these equations. 8.2. Continuous-time limitsFor the continuous time limit, we rewrite equations (8.4) in the form
$$
\begin{equation}
\begin{aligned} \, \nonumber &\frac{\kappa^m (x-\eta)}{\kappa^{m-1}(x)}\prod_j \frac{\sigma(\lambda_i^m(x)-\lambda_j^{m+1}(x))\sigma(\lambda_i^m(x)- \lambda_j^{m}(x-\eta)) \sigma(\lambda_i^m(x)-\lambda_j^{m-1}(x\,{+}\,\eta))} {\sigma(\lambda_i^m(x)-\lambda_j^{m-1}(x)) \sigma(\lambda_i^m(x)- \lambda_j^{m+1}(x-\eta))\sigma(\lambda_i^m(x)-\lambda_j^{m}(x\,{+}\,\eta))} \\ \nonumber &\quad=-1+\kappa^m(x)\kappa^m (x-\eta)\prod_j \frac{\sigma(\lambda_i^m(x)-\lambda_j^{m+1}(x+\eta)) \sigma(\lambda_i^m(x)- \lambda_j^{m}(x-\eta))}{\sigma(\lambda_i^m(x)- \lambda_j^{m}(x+\eta)) \sigma(\lambda_i^m(x)-\lambda_j^{m+1}(x-\eta))} \\ &\quad\quad\,+\kappa^{m-1}(x-\eta)\kappa^m (x-\eta)\prod_j \frac{\sigma(\lambda_i^m(x)-\lambda_j^{m+1}(x))\sigma(\lambda_i^m(x)- \lambda_j^{m-1}(x-\eta))}{\sigma(\lambda_i^m(x)- \lambda_j^{m-1}(x))\sigma(\lambda_i^m(x)-\lambda_j^{m+1}(x-\eta))} \end{aligned}
\end{equation}
\tag{8.14}
$$
and assume that the $\lambda_j^m$ and $\rho^m$ vary smoothly with time, that is,
$$
\begin{equation*}
\begin{aligned} \, \lambda_i^{m\pm 1}(x)&=\lambda_i(x)\pm \varepsilon \dot\lambda_i(x)+ \frac{1}{2}\varepsilon^2 \ddot\lambda_i(x)+O(\varepsilon^3), \\ \rho^{m\pm 1}(x)&=\rho(x)\pm \varepsilon \dot \rho(x)+O(\varepsilon^2). \end{aligned}
\end{equation*}
\notag
$$
The limit is straightforward. We expand the equation in powers of $\varepsilon$ as $\varepsilon \to 0$. If $\kappa^2 \ne 1$, then the first non-vanishing order is $\varepsilon$, and we obtain the equations
$$
\begin{equation}
\begin{aligned} \, \nonumber &\ddot\lambda_i(x)+\sum_j\bigl[\dot\lambda_i(x)\dot\lambda_j(x-\eta) \zeta\bigl(\lambda_i(x)-\lambda_j(x-\eta)\bigr)\nonumber\\ &\qquad\qquad\qquad\qquad\qquad+ \dot\lambda_i(x)\dot\lambda_j(x+\eta) \zeta\bigl(\lambda_i(x)-\lambda_j(x+\eta)\bigr)\bigr] \nonumber \\ &\qquad-2\sum_{j\ne i}\dot \lambda_i(x) \dot\lambda_j(x) \zeta\bigl(\lambda_i(x)-\lambda_j(x)\bigr)+ \bigl(c(x-\eta)-c(x)\bigr)\dot\lambda_i(x)=0, \end{aligned}
\end{equation}
\tag{8.15}
$$
where
$$
\begin{equation*}
c(x)=\frac{\dot \rho (x+\eta)}{\rho (x+\eta)}- \frac{\dot\rho(x)}{\rho(x)}\,.
\end{equation*}
\notag
$$
Summing equations (8.15) over all $i$ and taking condition (8.2) into account we find that $c(x)$ is expressed in terms of the $\lambda_i$ as follows:
$$
\begin{equation}
c(x)=\biggl(\,\sum_k\dot \lambda_k(x)\biggr)^{-1}\sum_{i,j}\dot\lambda_i(x) \dot\lambda_j(x+\eta)\zeta\bigl(\lambda_i(x)-\lambda_j(x+\eta)\bigr).
\end{equation}
\tag{8.16}
$$
Equations (8.15) and (8.16) were obtained in [28] as the equations of motion for the field analogue of the RS model. If $\kappa^2=1$, then the order of $\varepsilon$ gives the identity $0=0$, and one has to expand up to the order of $\varepsilon^2$. In this way it is possible to obtain a field analogue of equations (6.4). We do not present it here because it is rather bulky. Like in the previous section, another continuum limit is possible. We introduce the lattice fields $\varphi_i^m(x)$ and $\omega^m(x)$ by setting
$$
\begin{equation}
\lambda_i^m(x)=\varphi_i^m(x+m\eta)\quad\text{and} \quad \rho^m(x)=\omega^m (x+m\eta),
\end{equation}
\tag{8.17}
$$
and assume that these new fields have a smooth continuous-time limit in the following sense:
$$
\begin{equation*}
\varphi_i^{m\pm 1}(x)=\varphi_i(x)\pm \varepsilon\dot\varphi_i(x)+ \frac{1}{2}\,\varepsilon^2 \ddot\varphi_i(x)+O(\varepsilon^3)
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\omega^{m\pm 1}(x)=\omega(x)\pm \varepsilon \dot\omega(x)+O(\varepsilon^2).
\end{equation*}
\notag
$$
In terms of the new fields equations (8.14) read
$$
\begin{equation}
\begin{aligned} \, \nonumber &\frac{\kappa^m (x-\eta)}{\kappa^{m-1}(x)}\kern-1pt\prod_j\kern-1pt \frac{\sigma(\varphi_i^m(x)-\kern-1pt\varphi_j^{m+1}(x+\eta)) \sigma(\varphi_i^m(x)-\kern-1pt \varphi_j^{m}(x-\eta)) \sigma(\varphi_i^m(x)- \kern-1pt \varphi_j^{m-1}(x))} {\sigma(\varphi_i^m(x)-\kern-1pt\varphi_j^{m}(x+\eta)) \sigma(\varphi_i^m(x)-\kern-1pt \varphi_j^{m-1}(x-\eta)) \sigma(\varphi_i^m(x)-\kern-1pt\varphi_j^{m+1}(x))} \\ \nonumber &=-1+\kappa^m(x)\kappa^m (x-\eta)\prod_j \frac{\sigma(\varphi_i^m(x)-\varphi_j^{m+1}(x+2\eta))\sigma(\varphi_i^m(x)- \varphi_j^{m}(x-\eta))}{\sigma(\varphi_i^m(x)-\varphi_j^{m}(x+\eta)) \sigma(\varphi_i^m(x)-\varphi_j^{m+1}(x))} \\ &\qquad\;\; +\kappa^{m-1}(x-\eta)\kappa^m (x-\eta)\nonumber\\ &\qquad\qquad\qquad\times\prod_j \frac{\sigma(\varphi_i^m(x)-\varphi_j^{m-1}(x-2\eta)) \sigma(\varphi_i^m(x)- \varphi_j^{m+1}(x+\eta))}{\sigma(\varphi_i^m(x)- \varphi_j^{m-1}(x-\eta)) \sigma(\varphi_i^m(x)-\varphi_j^{m+1}(x))}\,. \end{aligned}
\end{equation}
\tag{8.18}
$$
The limit $\varepsilon \to 0$ exists if $\kappa=\varepsilon$. Then in the order $\varepsilon$ we obtain the equations
$$
\begin{equation}
\begin{aligned} \, \nonumber &\ddot\varphi_i(x)+\sum_j \bigl[\dot \varphi_i(x)\dot \varphi_j(x-\eta) \zeta \bigl(\varphi_i(x)-\varphi_j(x-\eta)\bigr)\nonumber \\ &\qquad\qquad\qquad\qquad+ \dot \varphi_i(x) \dot\varphi_j(x+\eta) \zeta \bigl(\varphi_i(x)- \varphi_j(x+\eta)\bigr)\bigr] \nonumber \\ \nonumber &\qquad-2\sum_{j\ne i}\dot \varphi_i(x) \dot\varphi_j(x) \zeta\bigl(\varphi_i(x)-\varphi_j(x)\bigr)-\partial_t\log (w(x))\dot\varphi_i(x) \\ &\qquad+w(x)w(x+\eta)G_i^++w(x)w(x-\eta)G_i^-=0, \end{aligned}
\end{equation}
\tag{8.19}
$$
where
$$
\begin{equation}
w(x)=\frac{\omega(x+\eta)\omega(x-\eta)}{\omega^2(x)}
\end{equation}
\tag{8.20}
$$
and
$$
\begin{equation}
\begin{aligned} \, \nonumber G_i^+&=\frac{\sigma(\varphi_i(x)-\varphi_i(x+2\eta))\sigma(\varphi_i(x)- \varphi_i(x-\eta))}{\sigma(\varphi_i(x)-\varphi_i(x+\eta))} \\ &\qquad\times\prod_{j\ne i}\frac{\sigma(\varphi_i(x)-\varphi_j(x+2\eta)) \sigma(\varphi_i(x)-\varphi_j(x-\eta))}{\sigma(\varphi_i(x)-\varphi_j(x+\eta)) \sigma(\varphi_i(x)-\varphi_j(x))}\,, \end{aligned}
\end{equation}
\tag{8.21}
$$
and $G_i^-$ differs from $G_i^+$ by the change $\eta \to -\eta$. Summing equations (8.19) over all $i$, we obtain the equation
$$
\begin{equation}
\partial_t \log(w(x)) \sum_i \dot \varphi_i(x)= F(x)+w(x)w(x+\eta)\sum_i G_i^+ +w(x)w(x-\eta)\sum_i G_i^-,
\end{equation}
\tag{8.22}
$$
where
$$
\begin{equation}
\begin{aligned} \, \nonumber F(x)&=\sum_{i,j}\bigl[\dot \varphi_i (x)\dot \varphi_j (x+\eta)\zeta \bigl( \varphi_i (x)-\varphi_j(x+\eta)\bigr) \\ &\qquad+\dot\varphi_i(x)\dot\varphi_j(x-\eta)\zeta\bigl(\varphi_i(x)- \varphi_j(x-\eta)\bigr)\bigr]. \end{aligned}
\end{equation}
\tag{8.23}
$$
Equations (8.19), together with (8.22), form a system of $N+1$ differential equations for the $N+1$ fields $\varphi_j(x)=\varphi_j(x,t)$ ($j=1,\dots,N$) and $w(x)=w(x,t)$. In contrast to (8.15), where the extra field was not dynamical and could be excluded from the equations of motion, in the present case the extra field $w$ is dynamical. Equations (8.19) and (8.22) provide the field extension of the deformed RS system (1.4).
9. Concluding remarks and open problemsIn this paper we have found the complete set of integrals of motion for the deformed RS system with equations of motion (1.4). This provides sufficient evidence for the integrability of the system. Our method is based on the fact that the deformed RS system is equivalent to the dynamical system for pairs of particles in the standard RS model (with an even number of particles) moving as single wholes so that the distance between particles in each pair is equal to $\eta$, the inverse ‘velocity of light’ in the RS ($=$ relativistic CM) model. Such pairs are preserved by only ‘one-half’ of the higher Hamiltonian flows, so we consider only $\mathsf{H}_k^-$-flows and put the time variables associated with the $\mathsf{H}_k^+$-flows equal to zero. The configurations in the whole of the phase space $\mathcal{F}$ when particles stick together in pairs form a half-dimensional subspace $\mathcal{P}\subset \mathcal{F}$, and we have proved that this subspace is Lagrangian and invariant under all $\mathsf{H}_k^-$-flows. Then integrals of motion for the deformed RS system can be obtained by restricting the known RS integrals of motion to the subspace $\mathcal{P}$. This work has been done in our paper. In the $\eta \to 0$ limit (when the RS system reduces to the CM system) the particles in each pair merge into a single point. This singular limiting case was discussed in [34]. It is an interesting question whether any clusters of RS particles other than pairs are possible in this sense. For example, one can consider ‘strings’ of $M$ particles such that the coordinates of the particles in the $i$th string are
$$
\begin{equation*}
x_{Mi+1-\alpha}=X_i+(M-\alpha)\eta,\qquad \alpha=1,\dots,M,
\end{equation*}
\notag
$$
where $X_i$ is the coordinate of the string moving as a single whole. It is natural to ask whether some Hamiltonian flows of the RS model preserve such string structure. We must stress that the connection between the standard RS system and the deformed RS system is not trivial and has different aspects. On the one hand the latter is an extension of the former and includes it as a particular case because the equations of motion (1.4) differ from the equations of motion (1.2) of the RS system by the presence of additional terms. However, on the other hand the deformed RS system is contained in the RS system since it can be regarded as its reduction in the sense that the equations of motion (1.4) are obtained by restricting the RS dynamics to the subspace $\mathcal{P}$ of pairs. We have obtained integrable time discretization of the deformed RS system. Our method is based on the explicit form of the Bäcklund transformation which is obtained as equations connecting the dynamics of poles and zeros of double-Bloch solutions $\psi$ to the linear problem for the Toda lattice with constraint of type B. As in other systems of CM and RS type, the Bäcklund transformation is the transition from poles to zeros of the $\psi$-function, and this is interpreted as one step forward in discrete-time evolution. Possible continuum limits of the discrete-time equations obtained have also been discussed. One of them gives the equations of motion of the deformed RS system while another gives the equations of motion for poles of elliptic solutions to the semi-discrete BKP equation [18]. We have also shown that the discrete-time equations of motion for the deformed RS system describe the evolution of poles of elliptic solutions to the fully discrete BKP equation. In addition, by considering more general elliptic solutions of the latter (so-called elliptic families) we have obtained a field extension of the deformed RS model on a space-time lattice. We would like to mention that the discrete-time equations of motion of the RS system (1.11) coincide mysteriously with the nested Bethe ansatz equations arising in the theory of quantum integrable systems with elliptic $R$-matrix. It is still not clear whether this is just a coincidence or this fact has some profound reasons. In this connection it is natural to ask whether the discrete-time equations (1.12) for the deformed RS system have any relation to quantum integrable systems. Namely, the question is whether or not there exists any quantum integrable system solved by the Bethe ansatz or any other method whose Bethe-like equations would be of the form (1.12). Finally, let us list some open problems which arise in connection with the deformed RS system. First, it is important to answer the question whether or not the deformed RS system is Hamiltonian. A related problem is the quantization of the deformed RS system. Second, it would be highly desirable to find commutation representations for the equations of motion (1.4) and (1.12) such as Lax’s representation or Manakov’s triple representation [35]. It is the latter that is known to exist for equations (1.6) obtained from (1.4) in the limit as $\eta \to 0$. This is why it is natural to conjecture that Manakov’s triple representation exists for equations (1.4) for all $\eta \ne 0$. We hope to discuss these problems elsewhere. 10. Appendix A: Weierstrass functionsIn this appendix we present the definition and main properties of the Weierstrass functions, namely, the $\sigma$-function, the $\zeta$-function, and the $\wp$-function, which are widely used in the main text. Let $\omega$ and $\omega '$ be complex numbers such that $\operatorname{Im}(\omega'/\omega)>0$. The Weierstrass $\sigma$-function with quasi-periods $2\omega$ and $2\omega'$ is defined by the following infinite product over the lattice $2\omega m+2\omega'm'$, $m,m'\in \mathbb{Z}$:
$$
\begin{equation}
\begin{gathered} \, \sigma(x)=\sigma(x\mid\omega,\omega')= x\prod_{s\ne 0}\biggl(1-\frac{x}{s}\biggr) \exp\biggl(\frac{x}{s}+\frac{x^2}{2s^2}\biggr), \\ \nonumber s=2\omega m+2\omega' m',\quad m,m'\in\mathbb{Z}. \end{gathered}
\end{equation}
\tag{10.1}
$$
It is an odd entire quasiperiodic function in the complex plane. As $x\to 0$, The monodromy properties of the $\sigma$-function under shifts by the quasi-periods are as follows:
$$
\begin{equation}
\begin{aligned} \, \sigma(x+2\omega)&=-\exp\bigl(2\zeta(\omega)(x+\omega)\bigr)\sigma(x), \\ \sigma(x+2\omega')&=-\exp\bigl(2\zeta(\omega')(x+\omega')\bigr)\sigma(x). \end{aligned}
\end{equation}
\tag{10.3}
$$
Here $\zeta (x)$ is the Weierstrass $\zeta$-function defined by The monodromy properties imply that
$$
\begin{equation*}
f(x)=\prod_{\alpha=1}^M \frac{\sigma(x-a_\alpha)}{\sigma(x-b_\alpha)}\,,\qquad \sum_{\alpha=1}^M(a_\alpha-b_\alpha)=0,
\end{equation*}
\notag
$$
is a doubly periodic function with periods $2\omega$ and $2\omega'$ (an elliptic function). The Weierstrass $\zeta$-function can be represented as a sum over the lattice as follows:
$$
\begin{equation}
\zeta(x)=\frac{1}{x}+\sum_{s\ne 0} \biggl(\frac{1}{x-s}+\frac{1}{s}+\frac{x}{s^2}\biggr),\qquad s=2\omega m+2\omega' m',\quad m,m'\in\mathbb{Z}.
\end{equation}
\tag{10.5}
$$
It is an odd function with first-order poles at the points of the lattice. As $x\to 0$, If the argument is shifted by any quasi-period, then the $\zeta$-function is transformed as follows:
$$
\begin{equation}
\begin{aligned} \, \zeta(x+2\omega)&=\zeta(x)+\zeta(\omega), \\ \zeta(x+2\omega')&=\zeta(x)+\zeta(\omega'). \end{aligned}
\end{equation}
\tag{10.7}
$$
These quantities $\zeta(\omega)$ and $\zeta(\omega')$ are related by the identity $2\omega'\zeta(\omega)-2\omega\zeta(\omega')=\pi i$. The transformation properties (10.7) imply that
$$
\begin{equation*}
g(x)=\sum_{\alpha =1}^M A_\alpha \zeta (x-a_\alpha),\qquad \sum_{\alpha =1}^M A_\alpha=0,
\end{equation*}
\notag
$$
is an elliptic function. The Weierstrass $\wp$-function is defined by $\wp(x)=-\zeta'(x)$. It can be represented as a sum over the lattice as follows:
$$
\begin{equation}
\wp(x)=\frac{1}{x^2}+\sum_{s\ne 0} \biggl(\frac{1}{(x-s)^2}- \frac{1}{s^2}\biggr),\qquad s=2\omega m+2\omega'm',\quad m,m'\in\mathbb{Z}.
\end{equation}
\tag{10.8}
$$
It is an even doubly periodic function with periods $2\omega$ and $2\omega'$ and with second-order poles at the lattice points $s=2\omega m+2\omega' m'$, where $m$ and $m'$ are integers. As $x\to 0$, The Weierstrass functions obey many non-trivial identities. Here we present the two that are necessary for the calculations leading to equation (6.4):
$$
\begin{equation}
\zeta(x+\eta)+\zeta (x-\eta)-2\zeta (x) = \frac{\wp '(x)}{\wp (x)-\wp(\eta)}\,,
\end{equation}
\tag{10.9}
$$
$$
\begin{equation}
\wp (x+\eta)-\wp (x-\eta) = -\frac{\wp'(x)\wp'(\eta)}{(\wp(x)-\wp(\eta))^2}\,.
\end{equation}
\tag{10.10}
$$
The proof is standard. Both sides are elliptic functions of $x$, and the singular terms on both sides coincide. Therefore, the difference between the left- and right-hand sides is a constant, which can be found by setting $x$ equal to a certain special value.
11. Appendix B: The proof of Lemma 4.1We set
$$
\begin{equation}
\begin{aligned} \, \nonumber F_m^{\pm}&=\sum_{\substack{\mathcal{I}\subset \mathcal{N}'\\ |\mathcal{I}|=m}}\,\prod_{i\in \mathcal{I}}\, \prod_{\ell \in \mathcal{N}'\setminus \mathcal{I}}U^{\pm}(X_{i\ell}) \\ &=\sum_{\substack{\mathcal{I}\subset \mathcal{N}'\\ |\mathcal{I}|=m}}\, \prod_{i\in \mathcal{I}}\, \prod_{\ell \in \mathcal{N}'\setminus \mathcal{I}} \frac{\sigma(X_{i\ell}\pm 2\eta)\sigma(X_{i\ell}\mp \eta)} {\sigma(X_{i\ell}\pm \eta)\,\sigma(X_{i\ell})} \end{aligned}
\end{equation}
\tag{11.1}
$$
and consider the function $f_m=F_m^{+}-F_m^{-}$. It is a symmetric function of the variables $X_j$, $j\in \mathcal{N}'$. It is easy to see that it is an elliptic function of each $X_j$. The statement of the lemma is that $f_m=0$ for all $m$. For $m=1$ we have
$$
\begin{equation}
f_1=\sum_{i\in \mathcal{N}'} \prod_{\ell\in\mathcal{N}'\atop \ell \ne i} \frac{\sigma(X_{i\ell}+ 2\eta)\,\sigma(X_{i\ell}- \eta)} {\sigma(X_{i\ell}+ \eta)\, \sigma(X_{i\ell})} -\sum_{i\in \mathcal{N}'}\, \prod_{\ell \in \mathcal{N}'\atop\ell \ne i} \frac{\sigma(X_{i\ell}- 2\eta)\,\sigma(X_{i\ell}+\eta)} {\sigma(X_{i\ell}-\eta)\,\sigma(X_{i\ell})}=0,
\end{equation}
\tag{11.2}
$$
since this expression is proportional to the sum of residues of the elliptic function
$$
\begin{equation*}
f(X)=\prod_{\ell \in \mathcal{N}'} \frac{\sigma(X-X_\ell+2\eta)\,\sigma(X-X_\ell-\eta)} {\sigma(X-X_\ell+\eta)\,\sigma(X-X_\ell)}.
\end{equation*}
\notag
$$
We are going to prove that $f_m=0$ for all $m$ by induction. Assume that $f_m=0$ for some $m$; we show that this is also true for $m\to m+1$. In view of symmetry it is enough to consider $f_m$ as a function of $X_1$ (without loss of generality we assume that $\mathcal{N}'\ni 1$). The possible poles of this function are first-order poles at $X_1=X_j$ and $X_1=X_j\pm\eta$. Let us prove that the residues at these poles actually vanish. For the poles at $X_1=X_j$ this is particularly simple because it is not difficult to see that even without the induction assumption. Consider the pole at $X_1=X_2+\eta$ (again, without loss of generality we can assume that $\mathcal{N}'\ni 2$). We introduce the shorthand notation
$$
\begin{equation*}
\mathcal{N}'_{1}=\mathcal{N}'\setminus \{1\},\quad \mathcal{N}'_{2}=\mathcal{N}'\setminus \{2\},\quad\text{and}\quad \mathcal{N}'_{12}=\mathcal{N}'\setminus \{1,2\}.
\end{equation*}
\notag
$$
Then we have
$$
\begin{equation}
\begin{aligned} \, \nonumber \operatorname*{res}_{X_1=X_2+\eta} f_m &=\sigma(2\eta) \sum_{\mathcal{I}\subseteq \mathcal{N}'_{12}}\, \prod_{\ell \in \mathcal{N}'_{12}\setminus \mathcal{I}} U^-(X_{1\ell})\prod_{i\in \mathcal{I}}\, \prod_{\ell \in \mathcal{N}'_{1} \setminus \mathcal{I}} U^-(X_{i\ell}) \\ &\qquad-\sigma(2\eta)\sum_{\mathcal{I}\subseteq \mathcal{N}'_{12}}\, \prod_{\ell \in \mathcal{N}'_{12}\setminus \mathcal{I}} U^+(X_{2\ell})\, \prod_{i\in \mathcal{I}}\, \prod_{\ell \in \mathcal{N}'_{2} \setminus \mathcal{I}} U^+(X_{i\ell}), \end{aligned}
\end{equation}
\tag{11.3}
$$
where $|\mathcal{I}|=m-1$. Since $X_1=X_2+\eta$, we have $U^+(X_{2\ell})=U^-(X_{1\ell})$. After simple transformations of the products, we can represent (11.3) in the form
$$
\begin{equation}
\begin{aligned} \, \nonumber \operatorname*{res}_{X_1=X_2+\eta} f_m&=\sigma(2\eta) \prod_{\ell \in \mathcal{N}'_{12}} U^-(X_{1\ell})\biggl[\,\sum_{\mathcal{I}\subseteq \mathcal{N}'_{12}}\, \prod_{i\in \mathcal{I}}\, \prod_{\ell \in \mathcal{N}'_{12} \setminus \mathcal{I}} U^-(X_{i\ell}) \\ &\qquad-\sum_{\mathcal{I}\subseteq \mathcal{N}'_{12}}\, \prod_{i\in \mathcal{I}}\, \prod_{\ell \in \mathcal{N}'_{12} \setminus \mathcal{I}}U^+(X_{i\ell})\biggr]. \end{aligned}
\end{equation}
\tag{11.4}
$$
The expression in square brackets is just $f_{m-1}$, which is zero by the induction assumption. Therefore, for all $m$. The pole at $X_1=X_2-\eta$ and the poles at $X_1=X_j\pm \eta$ are considered in a similar way. We have shown that the elliptic function $f_m$ is regular as a function of $X_1$. Therefore, it does not depend on $X_1$. By symmetry this function is a constant independent of all the $X_j$. To find this constant we can put $X_{j}=\varepsilon j$ and let $\varepsilon \to 0$. It is easy to see that after this substitution $f_m$ becomes an odd function of $\varepsilon$, so in the expansion as $\varepsilon \to 0$ the constant term $\propto \varepsilon^0$ vanishes. This means that the constant is equal to zero. I thank A. Marshakov and V. Prokofev for discussions.
Citation in format AMSBIB
\Bibitem{Zab23} |
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