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Algebraic-geometry approach to construction of semi-Hamiltonian systems of hydrodynamic type E. V. Glukhovabc, O. I. Mokhovabc a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow b Lomonosov Moscow State University, Faculty of Mechanics and Mathematics c Moscow Center for Fundamental and Applied Mathematics
Russian version:
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 2023, Volume 87, Issue 6, DOI: https://doi.org/10.4213/im9303 
Bibliographic databases:
    § 1. IntroductionRecall that a one-dimensional system of hydrodynamic type is an evolutionary system of quasilinear partial differential equations of the first order on the functions $u^k(q,t)$, $k=1,\dots,n$, of two independent variables $q$ and $t$, that is, a system of differential equations of the form
$$
\begin{equation}
u^k_t=\sum_{i=1}^n v^k_i(u) u^i_q, \qquad k=1,\dots,n,
\end{equation}
\tag{1.1}
$$
where the coefficients $v^k_i$ depend on $u=(u^1,\dots,u^n)$ (see [1], [2]). Here we use standard notation for partial derivatives: $u^i_t=\partial_t u^i$, $u^i_q=\partial_q u^i$. It is evident that the coefficients $v^k_i(u)$, $1 \leqslant i,k \leqslant n$, of system (1.1) are transformed as a tensor of type $(1,1)$ (an affinor) under non-degenerate changes $w= w(u)$. We will consider the case of diagonalizable affinors $v^k_i(u)$ (in other words, this is the case of diagonalizable one-dimensional systems of hydrodynamic type). The property of an affinor being diagonalizable by a change of coordinates in a certain neighbourhood is tensorial: the equality to zero of some tensor constructed from the affinor under consideration is a necessary and sufficient condition for the affinor to be diagonalizable by a change of coordinates in a neighbourhood for all affinors that are diagonalizable at each point of this neighbourhood. Such a tensor is the Haantjes tensor defined by an arbitrary affinor [3] (see also [4]). Thus, there is an efficient criterion to determine whether a one-dimensional system of hydrodynamic type is diagonalizable. The Hamiltonian formalism for systems of hydrodynamic type of general form was proposed by Dubrovin and Novikov in [1], where it was proved that the non-degenerate local Poisson brackets of hydrodynamic type introduced in this paper (the Dubrovin–Novikov brackets) are generated by flat metrics. Developing this differential-geometric Hamiltonian approach of Dubrovin and Novikov for diagonal systems of hydrodynamic type with distinct eigenvalues $\lambda_i(u)$, $\lambda_i(u)\neq\lambda_j(u)$, if $i\neq j$, $1 \leqslant i,j \leqslant n$, Tsarev [2] identified a class of semi-Hamiltonian diagonal systems of hydrodynamic type with distinct eigenvalues, which are specified by conditions (the semi-Hamiltonian conditions)
$$
\begin{equation}
\partial_j\biggl(\frac{\partial_k \lambda_i}{\lambda_k-\lambda_i}\biggr) = \partial_k\biggl(\frac{\partial_j \lambda_i}{\lambda_j-\lambda_i}\biggr), \qquad i,j,k\text{ are distinct},
\end{equation}
\tag{1.2}
$$
and proved that semi-Hamiltonian diagonal systems of hydrodynamic type are integrable by the generalized hodograph method developed in [2]. To each semi-Hamiltonian diagonal system of hydrodynamic type there corresponds a diagonal non-degenerate metric
$$
\begin{equation*}
g(u)=\operatorname{diag}\bigl(H_1^2(u),\dots,H_n^2(u)\bigr),
\end{equation*}
\notag
$$
satisfying the conditions
$$
\begin{equation}
\Gamma^i_{ij}=\partial_j (\ln H_i)= \frac{\partial_j \lambda_i}{\lambda_j-\lambda_i}, \qquad i\neq j.
\end{equation}
\tag{1.3}
$$
The consistency of the system of differential equations (1.3) for the Lamé coefficients $H_i(u)$, $i=1,\dots,n$, of the diagonal metric $g(u)$ is equivalent to the semi-Hamiltonian conditions (1.2). The conditions for a system to be semi-Hamiltonian impose the following conditions on the curvature tensor of the corresponding diagonal metric $g(u)$ in the coordinates under consideration: This condition distinguishes the class of spaces of diagonal curvature. The space of diagonal curvature is a Riemannian or pseudo-Riemannian space in which there exist orthogonal coordinates such that in these coordinates condition (1.4) is satisfied for the metric. Thus, to semi-Hamiltonian systems there correspond diagonal metrics with the condition of diagonal curvature (1.4). Moreover, for any diagonal metric with the condition of diagonal curvature (1.4), a linear system of differential equations (1.3) on the functions $\lambda_i(u)$, $i=1,\dots,n$, is consistent and any of its solutions, for which all $\lambda_i(u)$, $i=1,\dots,n$, are distinct, define semi-Hamiltonian diagonal systems of hydrodynamic type. Thus, to diagonal metrics with the condition of diagonal curvature (1.4) there correspond semi-Hamiltonian systems of hydrodynamic type. Let us note here an interesting fact that for the metric of the space of diagonal curvature, the diagonal curvature condition (1.4) may not be satisfied in arbitrary orthogonal coordinates (see [5]). But it is easy to show that a semi-Hamiltonian diagonal system of hydrodynamic type, under non-degenerate changes of variables $w=w(u)$ that leave it diagonal, preserves not only the eigenvalues of the affinor, but also the semi-Hamiltonian condition. Moreover, in [6], for diagonalizable affinors with distinct eigenvalues, a tensor of the semi-Hamiltonian property was constructed, which is identically equal to zero if and only if the corresponding diagonal system of hydrodynamic type with distinct eigenvalues is semi-Hamiltonian, that is, the semi-Hamiltonian condition is tensorial. Thus, checking the conditions that a hydrodynamic type system (1.1) is a semi-Hamiltonian diagonalizable system is not difficult: it is necessary to check that the eigenvalues of the affinor are distinct, and to check the tensor conditions for diagonalizability and the semi-Hamiltonian property for the affinor. However, the construction of semi-Hamiltonian diagonal systems of hydrodynamic type is a nontrivial nonlinear problem. In this paper, this problem is solved using algebraic-geometric methods. Semi-Hamiltonian diagonal systems of hydrodynamic type possess a continuum set of linearly independent hydrodynamic integrals and a continuum set of linearly independent hydrodynamic symmetries, which are parameterized by $n$ arbitrary functions of one variable [2]. Moreover, for generic systems of hydrodynamic type (or, in other words, for generic affinors), the converse statement also holds: if a system of hydrodynamic type (1.1) with a generic affinor with distinct eigenvalues possesses a continuum set linearly independent hydrodynamic integrals (or a continuum set of linearly independent hydrodynamic symmetries), which are parameterized by $n$ arbitrary functions of one variable, then this system of hydrodynamic type is diagonalizable (see [6]), and therefore semi-Hamiltonian, since Tsarev [2] proved that diagonal systems of hydrodynamic type with distinct eigenvalues are semi-Hamiltonian if and only if they possess a continuum set of linearly independent hydrodynamic integrals (or a continuum set of linearly independent hydrodynamic symmetries), which are parameterized by $n$ arbitrary functions of one variable. By hydrodynamic integrals we mean functionals of the form with densities $P(u)$ independent on derivatives of the functions $u$, which are the first integrals of the system: $(I[u])_t=0$.The Hamiltonian geometry of semi-Hamiltonian systems of hydrodynamic type is of particular interest. In important special cases of diagonal metrics with the condition of diagonal curvature (1.4) the Hamiltonian structures of semi-Hamiltonian systems are given by local Dubrovin–Novikov brackets [1] (for diagonal flat metrics), nonlocal Mokhov–Ferapontov brackets [7] (for diagonal metrics of constant curvature) and nonlocal Ferapontov brackets [8] (for diagonal metrics of submanifolds in pseudo-Euclidean spaces with flat normal bundle and holonomic net of curvature lines in the coordinates of the curvature lines). Note that systems of hydrodynamic type that possess two such compatible Hamiltonian structures (in the general case, two compatible nonlocal Ferapontov brackets) are necessarily diagonalizable if the pair of metrics of these Hamiltonian structures has distinct eigenvalues (see [9]–[11]). Diagonal flat metrics, diagonal metrics of constant curvature and diagonal metrics of submanifolds in pseudo-Euclidean spaces with flat normal bundle and holonomic net of curvature lines in the coordinates of the curvature lines are always diagonal metrics with the condition of diagonal curvature (1.4). Diagonal metrics with the condition of diagonal curvature (1.4) are described by a nonlinear system of partial differential equations (the Darboux equations), which is integrable by the inverse scattering method. In the papers of Zakharov [12], [13] it is shown that nonlinear partial differential equations describing the diagonal metrics of all the classes listed above are also integrable by the inverse scattering method. For the explicit construction of diagonal metrics of these important special classes, algebraic-geometric constructions were proposed and developed by Krichever [14] (for diagonal flat metrics), Berdinsky and Rybnikov [15] (for diagonal metrics of constant curvature) and the authors of the present paper [16], [17] (for diagonal metrics of submanifolds in pseudo-Euclidean spaces with flat normal bundle and holonomic net of curvature lines in the coordinates of the curvature lines and for diagonal flat metrics of a special form). All metrics obtained by these algebraic-geometric methods are diagonal metrics with diagonal curvature and correspond to some semi-Hamiltonian diagonal systems of hydrodynamic type. In the present paper, we propose a generalization of the constructions mentioned above for the construction of semi-Hamiltonian diagonal systems of hydrodynamic type from algebraic-geometric data. In addition, we will obtain expressions for hydrodynamic integrals and hydrodynamic symmetries of such systems in terms of Baker–Akhiezer functions on a smooth algebraic curve. This algebraic-geometric method for constructing semi-Hamiltonian diagonal systems of hydrodynamic type, as well as their hydrodynamic integrals and hydrodynamic symmetries, allows to find explicit formulas in $\theta$-functions of a certain algebraic curve. When an algebraic curve degenerates, this method allows to obtain all formulas in elementary functions. In the case of Krichever’s construction for diagonal flat metrics, such an approach was proposed by Mironov and Taimanov in [18], where the limiting case of Krichever’s construction for orthogonal coordinates in flat space is considered, when the algebraic curve becomes singular and reducible, and all its irreducible components are smooth rational complex curves. In this case, all the resulting formulas are expressed in elementary functions. For more general algebraic-geometric constructions, similar degenerations were also studied in [15]–[17]. Note also that the case of non-diagonalizable affinors and corresponding non-diagonalizable systems of hydrodynamic type is of particular great interest. The affinors of some of the most important integrable systems of hydrodynamic type associated with the associativity equations of two-dimensional topological field theories and equations of Monge–Ampère type are non-diagonalizable (see [19]–[21]). Such non-diagonalizable integrable systems of hydrodynamic type have quite different nature of integrability and geometry, including Hamiltonian geometry (see [19]–[27]). Section 2 provides a construction for constructing diagonal metrics with diagonal curvature from algebraic-geometric data; in § 3, affinors of semi-Hamiltonian diagonal systems were constructed using algebraic-geometric data; in § 4 formulas for hydrodynamic integrals and hydrodynamic symmetries of algebraic-geometric semi-Hamiltonian systems are proved. § 2. Algebraic-geometric metrics of diagonal curvatureLet us introduce the algebraic-geometric data we need:
$$
\begin{equation*}
\bigl\{\Gamma,\, g;\, \{P_j,k_j^{-1}\}_{j=1}^n;\, R_1,\dots,R_{l+N};\, \gamma_1,\dots,\gamma_{g+l+N-1}\bigr\}.
\end{equation*}
\notag
$$
Consider a smooth algebraic curve $\Gamma$ of genus $g$, and choose three divisors on $\Gamma$:
$$
\begin{equation}
P=P_1+\dots+P_n,\quad\gamma=\gamma_1+\dots+\gamma_{g+l+N-1},\quad R=R_1+\dots+R_{l+N},
\end{equation}
\tag{2.1}
$$
where all points of the divisors are assumed to be distinct, $n$ is an arbitrary positive integer, $l$, $N$ are arbitrary nonnegative integers. In a neighbourhood of each point $P_i$ we introduce a local parameter $k_i^{-1}$, $k_i^{-1}(P_i)=0$. A multipoint Baker–Akhiezer function is a function $\psi(u,Q)$, $u=(u^1,\dots,u^n)$, $Q\in\Gamma$, with the following analytical properties (see [14]): 1) at each point $P_j$ the function $\psi$ has an essential singularity and an expansion in a neighbourhood of the form
$$
\begin{equation}
\psi (u,Q)=e^{k_ju^j}\biggl(\xi_{j,0}(u)+\frac{\xi_{j,1}(u)}{k_j}+ \cdots\biggr);
\end{equation}
\tag{2.2}
$$
2) the function $\psi(u,Q)$ in the variable $Q\in\Gamma$ is meromorphic outside points of the divisor $P$ and has simple poles at points of the divisor $\gamma$; 3) at points of the divisor $R$ the function takes constant real values: $\psi(u, R_{\alpha})=d_{\alpha}\in\mathbb{R}$, $\alpha=1,\dots,l+N$. Such a function exists and is unique, so if the vector of normalization constants $d = (d_1,\dots,d_{l+N})$ is equal to the zero vector, then $\psi(u,Q)\equiv 0$. A curve $\Gamma$ is said to be admissible if there is a holomorphic involution $\sigma\colon \Gamma\to\Gamma$, $\sigma^2(Q)=Q$, such that 1) there are exactly $2n + 2N$ fixed points of the involution: $P_1,\dots,P_n$, $R_{l+1},\dots, R_{l+N}$, and the remaining fixed points will be denoted by $Q_1,\dots,Q_{n+N}$; 2) in a neighbourhood of each point of the divisor $P$, the local parameters are odd with respect to the involution: $\sigma (k_i^{-1})=-k_i^{-1}$. For example, any hyperelliptic curve is admissible: if we choose a permutation of sheets as an involution, then the branch points will be fixed. A pair of divisors $\gamma$ and $R$ is called admissible if there is a meromorphic differential $\Omega$ on $\Gamma$ such that 1) the differential $\Omega$ has $n+2g+2l+2N-2$ first-order zeros at points of the divisors $P$, $\gamma$, $\sigma(\gamma)$; 2) the differential $\Omega$ has $n+2l+2N$ simple poles at the points $Q_1,\dots,Q_{n+N}$ and at the points $R_1,\dots, R_{l+N}$, $\sigma(R_1),\dots,\sigma(R_l)$. In order for the constructed algebraic-geometric functions to be real, we need to require the existence of some special anti-holomorphic involution $\tau\colon \Gamma\to\Gamma$ on the curve such that the points $P_1,\dots,P_n,Q_1,\dots,Q_{n+N}, R_1, \dots,R_{l+N}$ are fixed under the involution $\tau$, the local parameters in the neighbourhood of the points $P_i$ change anti-holomorphically: $\tau(k_i^{-1})=\overline{k_i^{-1}}$, and $\tau(\gamma)=\gamma$. For such an involution, the following is true: This involution is described in more detail in [14].If the curve $\Gamma$ is admissible, and the pair of divisors $\gamma$ and $R$ is also admissible, then the following theorem is true. Theorem 2.1 (see [16], [17]). The functions $x^k(u)=\psi(u, Q_k)$, $k = 1, \dots, n+ N$, define an $n$-dimensional submanifold of the flat $(n+N)$-dimensional space with the diagonal metric $(\operatorname{Res}_{Q_1}\Omega,\dots, \operatorname{Res}_{Q_{n+N}}\Omega)$ so that $u = (u^1, \dots, u^n)$ are orthogonal coordinates on the submanifold. In these orthogonal coordinates, the Lamé coefficients of the diagonal metric induced on the submanifold are given by the formulas where the constants $\varepsilon_i$ are determined from the expansion of the differential $\Omega$ in the neighbourhood of the points $P_i$: Let us note that in formula (2.4) only the functions $\xi _{i,0}(u)$ depend on the choice of the vector $d=(d_1,\dots,d_{l+N})$ normalizing the Baker–Akhiezer function, and the constants $\varepsilon_i$ are determined by algebraic-geometric data and the choice of the differential $\Omega$. In addition, using the same algebraic-geometric data and additional $n$ points $S = \{S_1,\dots,S_n\}$ on the curve $\Gamma$, we will construct another Baker–Akhiezer function $\widetilde{\psi}(u,Q)$ with the following analytical properties (see [14]): 1) at each point $P_j$ the function $\widetilde{\psi}$ has an essential singularity and an expansion in a neighbourhood of this point of the form
$$
\begin{equation}
\widetilde{\psi} (u,Q)=k_j e^{k_ju^j} \biggl(\widetilde{\xi}_{j,0}(u)+ \frac{\widetilde{\xi}_{j,1}(u)}{k_j}+\cdots\biggr);
\end{equation}
\tag{2.5}
$$
2) the function $\widetilde{\psi}(u,Q)$ in the variable $Q\in\Gamma$ is meromorphic outside points of the divisor $P$ and has simple poles at points of the divisor $\gamma$; 3) at points of the divisor $R$ the function takes constant values: $\widetilde{\psi}(u, R_{\alpha})=\widetilde{d}_{\alpha}$, $\alpha=1,\dots,l+N$, and at points $S_1,\dots, S_n$ has $n$ simple zeros: $\widetilde{\psi}(u,S_i)=0$, $i=1,\dots, n$. Such a function exists and is unique [14]. It is important for us that, due to the uniqueness, if $\widetilde{d}_{\alpha}=0$, $\alpha=1,\dots,l+N$, then such a function is identically equal to zero: $\widetilde{\psi}(u,Q)\equiv 0$. § 3. Algebraic-geometric semi-Hamiltonian diagonal systemsThe Baker–Akhiezer functions for given algebraic-geometric data form a vector space, which is parameterized by choosing a vector of normalization constants $d=(d_1,\dots,d_{l+N})$. In this space we choose two functions $\psi^{(a)}$ and $\psi^{(b)}$, defined by the vectors of normalization constants $a=(a_1,\dots,a_{l+N})$ and $b=(b_1,\dots,b_{l+N})$. Then the Baker-Akhiezer function specified by the vector $a+b$ is equal to the sum of the functions $\psi^{(a)}$ and $\psi^{(b)}$: The functions $\psi^{(a)}$, $\psi^{(b)}$ and $\psi^{(a+b)}$ define the embedding functions of three submanifolds of diagonal curvature and orthogonal coordinates with the condition of diagonal curvature (1.4) on these submanifolds:
$$
\begin{equation*}
x^i(u) = \psi^{(a)}(u, Q_i),\qquad y^i(u) = \psi^{(b)}(u, Q_i),\qquad z^i(u) = \psi^{(a+b)}(u, Q_i).
\end{equation*}
\notag
$$
Due to the linearity, the embedding functions $z(u) = (z^1 (u),\dots, z^{n+N}(u))$, which are given by the function $\psi^{(a+b)}$, are the sum of the embedding functions $x(u) = (x^1 (u),\dots, x^{n+N}(u))$ and $y(u) = (y^1 (u),\dots, y^{n+N}(u))$: Thus, the constructed submanifolds of diagonal curvature with orthogonal coordinates with the condition of diagonal curvature (1.4) on these submanifolds form a linear space, which is parameterized by a vector of normalization constants. If, after taking the sum of arbitrary submanifolds with orthogonal coordinates on them, we obtain a submanifold with orthogonal coordinates, then the Lamé coefficients change as follows:
$$
\begin{equation*}
\bigl( H_i^{(x+y)} \bigr)^2 = \bigl( \partial_i (x+y),\, \partial_i (x+y)\bigr) = \bigl( H_i^{(x)} \bigr)^2 + \bigl( H_i^{(y)} \bigr)^2 + 2 (\partial_i x, \partial_i y).
\end{equation*}
\notag
$$
On the other hand, according to Theorem 2.1, the algebraic-geometric Lamé coefficients are proportional to the first coefficients of the expansion of the corresponding Baker–Akhiezer function. And since the proportionality coefficient does not depend on the vector of normalization constants, the Lamé coefficients of the sum of two algebraic-geometric submanifolds of diagonal curvature are equal to the sum of the Lamé coefficients of each of these submanifolds:
$$
\begin{equation*}
\begin{gathered} \, H_i^{(x+y)}=H_i^{(x)}+H_i^{(y)}, \\ \bigl( H_i^{(x+y)} \bigr)^2 = \bigl( H_i^{(x)} \bigr)^2 + \bigl( H_i^{(y)} \bigr)^2 + 2 H_i^{(x)}H_i^{(y)}. \end{gathered}
\end{equation*}
\notag
$$
This means that for the constructed submanifolds it is true that And since $H_i^{(x)}=\sqrt{(\partial_i x, \partial_i x)}$, then condition (3.1) implies collinearity of the vectors $\partial_i x(u)$ and $\partial_i y(u)$ at each point. Therefore, there exists a set of functions $\lambda_i(u)$, $i=1,\dots,n$, such that Generally speaking, the functions $\lambda_i(u)$ depend on the choice of vectors $a$ and $b$. Consider the function
$$
\begin{equation*}
\psi^{(ab)}_i(u,Q)=\partial_i \psi^{(a)}(u,Q)- \lambda_i(u)\, \partial_i \psi^{(b)}(u,Q).
\end{equation*}
\notag
$$
This function is a Baker–Akhiezer function with essential singularities of the form $k_j e^{u^j k_j}$ (see (2.5)). In addition, this function is equal to zero at points $Q_1,\dots, Q_{n+N}$ due to (3.2) and at points $R_1,\dots, R_{l+N}$. From the uniqueness of functions of such type we find that $\psi^{(ab)}_i(u,Q)\equiv 0$. Since the function $\psi^{(ab)}_i$ is equal to zero, all the expansion coefficients in a neighbourhood of each point $P_j$ are equal to zero. If $j\neq i $, then in a neighbourhood of the point $P_j$ the function $\psi^{(ab)}_i$ is expanded as follows:
$$
\begin{equation*}
\psi^{(ab)}_i = k_j e^{u^j k_j}\bigl(\bigl(\partial_i \xi^{(a)}_{j,0}(u) - \lambda_i(u)\, \partial_i \xi^{(b)}_{j,0}(u)\bigr) k_j^{-1}+ \cdots\bigr).
\end{equation*}
\notag
$$
In the indicated expansion and below, the function $\xi^{(d)}_{j,0}(u)$ is the first coefficient of the expansion of the function $\psi^{(d)}$ in a neighbourhood of the point $P_j$. Knowing the expansion, we obtain the equation
$$
\begin{equation}
\partial_i \xi^{(a)}_{j,0} = \lambda_i\, \partial_i \xi^{(b)}_{j,0}, \qquad j\neq i.
\end{equation}
\tag{3.3}
$$
If $j = i $, then in a neighbourhood of the point $P_i$ the function $\psi^{(ab)}_i$ is expanded as follows:
$$
\begin{equation*}
\psi^{(ab)}_i = k_i\, e^{u^i k_i}\bigl(\bigl(\xi^{(a)}_{i,0}(u) - \lambda_i(u) \xi^{(b)}_{i,0}(u)\bigr)+ \cdots\bigr).
\end{equation*}
\notag
$$
Knowing the expansion, we get the equation Differentiate equation (3.4) with respect to $u^j$, $j\neq i$, we have
$$
\begin{equation*}
\partial_j \xi^{(a)}_{i,0} = \partial_j \lambda_i \, \xi^{(b)}_{i,0}+ \lambda_i \, \partial_j \xi^{(b)}_{i,0}.
\end{equation*}
\notag
$$
Applying equality (3.3) to the left-hand side, we get
$$
\begin{equation*}
\lambda_j \, \partial_j \xi^{(b)}_{i,0} = \partial_j \lambda_i \, \xi^{(b)}_{i,0}+ \lambda_i \, \partial_j \xi^{(b)}_{i,0}.
\end{equation*}
\notag
$$
From Theorem 2.1 it is known that $H^{(y)}_i= \varepsilon_i \xi^{(b)}_{i,0}$, $\varepsilon_i = \mathrm{const}$. Therefore,
$$
\begin{equation*}
\lambda_j \, \partial_j H^{(y)}_i = \partial_j \lambda_i \, H^{(y)}_i+ \lambda_i \, \partial_j H^{(y)}_i,\qquad \partial_j \lambda_i \, H^{(y)}_i = (\lambda_j-\lambda_i)\, \partial_j H^{(y)}_i.
\end{equation*}
\notag
$$
Now let $\lambda_i \neq \lambda_j$ for $i\neq j$. Then
$$
\begin{equation*}
\frac{\partial_j \lambda_i}{\lambda_j-\lambda_i} = \partial_j \bigl(\ln H^{(y)}_i\bigr).
\end{equation*}
\notag
$$
From the last equality it immediately follows that the diagonal system of hydrodynamic type $u^i_t=\lambda_i(u) u^i_q$, which corresponds to the diagonal metric of diagonal curvature with the Lamé coefficients $H_i^{(y)}$, is semi-Hamiltonian. Thus, the following theorem is proved. Theorem 3.1. The functions As a result, using algebraic-geometric data, we constructed a semi-Hamiltonian diagonal system of hydrodynamic type together with the corresponding diagonal metric $ds^2= \sum_{i=1}^n(H^{(y)}_i)^2\, (du^i)^2$ with the condition of diagonal curvature (1.4). In the papers of Glukhov and Mokhov [16], [17] it was proved that in the general case of arbitrary considered admissible algebraic-geometric data, such metrics are diagonal metrics of submanifolds of pseudo-Euclidean spaces with flat normal bundle and a holonomic net of curvature lines (in the coordinates of the curvature lines on the submanifold) (see Theorem 2.1 above). This Glukhov–Mokhov construction is a natural generalization of the well-known algebraic-geometric Krichever construction [14] for constructing flat diagonal metrics (the case $N = 0$) and the Berdinsky–Rybnikov construction [15] for constructing diagonal metrics of constant curvature (the case $N = 1$, $d_1=\dots=d_l=0$). Thus, taking into account [14]–[17] we obtain the following important consequences related to the metrics and Hamiltonian structures of the constructed semi-Hamiltonian diagonal systems of hydrodynamic type. Corollary 3.1. The algebraic-geometric semi-Hamiltonian diagonal systems constructed in Theorem 3.1 correspond to the following diagonal metrics with the condition of diagonal curvature: 1) for $N=0$ these metrics are flat; 2) for $N=1$ and $d_1=\dots=d_l=0$ these metrics are metrics of constant curvature; 3) in the general case of arbitrary admissible algebraic-geometric data, these metrics are metrics of submanifolds with flat normal bundle and a holonomic net of curvature lines (in the coordinates of the curvature lines). All these cases correspond to semi-Hamiltonian diagonal systems of hydrodynamic type, which possess Hamiltonian structures: for $N= 0$ these Hamiltonian structures are given by local Dubrovin–Novikov brackets (see [1] and [2]), for $N=1$, $d_1=\dots=d_l=0$, they are given by nonlocal Mokhov–Ferapontov brackets (see [7]) and in the general case of arbitrary admissible algebraic-geometric data these Hamiltonian structures are given by nonlocal Ferapontov brackets (see [8]). The rotation coefficients of a diagonal metric with Lamé coefficients $H_1(u),\dots, H_n(u)$ are the functions
$$
\begin{equation*}
\beta_{ij}(u)=\frac{\partial_i H_j(u)}{H_i(u)}, \qquad i \neq j.
\end{equation*}
\notag
$$
Two diagonal metrics are said to be related by the Combescure transformation if their rotation coefficients are the same. Corollary 3.2. The choice of different normalization constants $d_{\alpha}$, $\alpha=1,\dots,l+N$, in Theorem 2.1 gives metrics of diagonal curvature related by the Combescure transformation. Proof. Let us divide equality (3.3) by equality (3.4). Multiplying the left- and right-hand sides by $\varepsilon_j/\varepsilon_i$, we get This equality means the equality of the rotation coefficients of the corresponding metrics.
§ 4. Hydrodynamic integrals and hydrodynamic symmetries of algebraic-geometric semi-Hamiltonian systemsThe density of a hydrodynamic integral for a system of hydrodynamic type is a function $P(u)$ such that the functional does not depend on $t$ on any solution of the system of hydrodynamic type.According to [2], the family of densities of hydrodynamic integrals for semi-Hamiltonian systems is parameterized by $n$ univariable functions. All such integrals are described in the following lemma. Lemma 4.1 (Tsarev [2]). The function $P(u)$ is the density of a hydrodynamic integral of a semi-Hamiltonian system with the corresponding diagonal metric $g(u)=\operatorname{diag}(H_1^2(u),\dots,H_n^2(u))$ with the condition of diagonal curvature (1.4) if and only if where $\Gamma ^i_{ij}(u)= \partial_j(\ln H_i(u))$ are the Christoffel symbols of the metric. For the second derivatives of the Baker–Akhiezer functions, the following result is known (in our case the proof is completely similar). Lemma 4.2 (Krichever, [14]). The Baker–Akhiezer function $\psi(u,Q)$ satisfies the differential relation where $c_{ij}^i(u)=\partial_j \xi_{i,0}(u) / \xi_{i,0}(u)$. Applying these lemmas to the algebraic-geometric semi-Hamiltonian system of hydrodynamic type constructed in Theorem 3.1 and using Lemma 4.2 and property (2.3) of the anti-holomorphic involution $\tau$, we arrive at the following theorem. Theorem 4.1. For the semi-Hamiltonian diagonal system of hydrodynamic type constructed in Theorem 3.1, the function $\psi^{(b)}(u,Q_0)+\psi^{(b)}(u,\tau(Q_0))$ is the density of a real hydrodynamic integral of this system for an arbitrary point $Q_0$ on $\Gamma$. A system of hydrodynamic type $u_{T}^i = w^i_s(u) u^s_q$ is called hydrodynamic symmetry of a system of hydrodynamic type (1.1) if the corresponding flows commute, that is,
$$
\begin{equation*}
\bigl(v^i_s(u) u^s_q\bigr)_T = \bigl(w^i_s(u) u^s_q\bigr)_t.
\end{equation*}
\notag
$$
In the case of an arbitrary semi-Hamiltonian system of hydrodynamic type, the following lemma is true. Lemma 4.3 (Tsarev [2]). All hydrodynamic symmetries of a semi-Hamiltonian diagonal system of hydrodynamic type corresponding to the diagonal metric $ds^2=\sum_{i=1}^n(H_i)^2\, (du^i)^2$ with the condition of diagonal curvature (1.4), and only they are diagonal systems of hydrodynamic type, corresponding to the same metric (see condition (1.3)), and have the form where the metrics with the Lamé coefficients $H_i(u)$ and the Lamé coefficients $\widehat{H}_i(u)$ are related by the Combescure transformation. Applying this lemma to the algebraic-geometric semi-Hamiltonian systems of hydrodynamic type constructed in this paper and using equality (3.4), we arrive at the following result. Theorem 4.2. For the semi-Hamiltonian diagonal system of hydrodynamic type constructed in Theorem 3.1, a diagonal system of hydrodynamic type with diagonal elements is a hydrodynamic symmetry of this system for an arbitrary choice of the vector of normalization constants $\widehat{a}$. 
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