Korteweg–de Vries equation and hyperelliptic functions theory

We consider solutions of the hierarchy of the Korteweg-de Vries (KdV) equation $\dot{U} = 6UU' - U'''$ for the function $U = U(x,t)$ in the class of meromorphic functions $U(x,t) = u({\mathbf t})$ in $g$ variables ${\mathbf t}= (t_1, \ldots, t_{2g-1})$, where $t_1 = x$,  $t_3 = t$ and $u' = \frac{\partial}{\partial t_1}u,\, \dot{u} = \frac{\partial}{\partial t_3}u$. The dependence on higher times $t_{2g+1}, t_{2g+3}, \ldots$ is given by the formula
$$u({\mathbf t}) = u\left(t_1 - \sum\limits_{k=g+1}^\infty \alpha_{1k}t_{2k-1}, \ldots, t_{2g-1} - \sum\limits_{k=g+1}^\infty \alpha_{gk}t_{2k-1}\right),$$
where $\alpha_{ij}$ are constants. It leads to S.P.Novikov equations' system, that selects finite-gap (algebro-geometric) solutions of the KdV equation. For any non-singular hyperelliptic curve $V$ of genus $g$ an entire function $\sigma({\mathbf t};\lambda)$ in ${\mathbf t}= (t_1, \ldots, t_{2g-1})$ is uniquely defined, where $\lambda=(\lambda_4, \ldots ,\lambda_{4g+2})$ are the parameters of the curve $V$. In a neighborhood of point ${\mathbf t}=0$, the coefficients of the expansion of this function in a series in ${\mathbf t}$ are polynomials of $\lambda$. Logarithmic derivatives of order at least $2$ of the function $\sigma({\mathbf t};\lambda)$ determine meromorphic functions on the Jacobian $Jac(V)$ of the curve $V$. They are called basic hyperelliptic functions. In the case $g=1$ these are Weierstrass elliptic functions. The talk is devoted to the results that based on the theory of basic hyperelliptic functions (see. V.M. Buchstaber, V.Z. Enolskii, D.V. Leikin, Hyperelliptic Kleinian functions and applications, Solitons, Geometry and Topology: On the Crossroad, AMS Trans., 179:2, 1997, 1–33).
Set $u_{2k} = -2 \frac{\partial^2}{\partial t_1 \partial t_{2k-1}}\, \ln \sigma({\mathbf t};\lambda),\; k = 1,\ldots,g$.
The function $u_{2}$ gives a solution of the KdV hierarchy. We show that the KdV hierarchy, the integrals of flows of KdV and solutions of S.P.Novikov equations' system are effectively described in terms of the differential ring in $g$ functions $u_{2}, \ldots ,u_{2g}$, that is isomorphic to a ring of polynomials in $3g$ variables $u_{2}, u_{2}',u_{2}'',\ldots, u_{2g}, u_{2g}', u_{2g}''$. As a result we obtain $g$ integrable polynomial dynamical systems, determined by $g$ commuting flows in a $3g$-dimensional space $\mathbb{C}^{3g}$. This systems have $2g$ common polynomial integrals.
The case $g = 1$ and 2 is considered in detail in the paper “V. M. Buchstaber, Polynomial dynamical systems and the Korteweg–de Vries equation, Tr. Mat. Inst. Steklova, 2016, 294, 191–215”.