Integrable polynomial dynamical systems determined by algebraic curves

The best-known of the systems that will be discussed in the talk are the dynamical systems leading to the Korteweg–de Vries hierarchy (see papers by S. P. Novikov and his school). This systems are determined by hyperelliptic curves $V$ of genus $g>0$. Using the Abel–Jacobi map for the $g$-th symmetric power of the curve $V$, one can in an explicit form write down this solutions in multidimensional analogues of Weierstrass functions. They are the basic meromorphic functions on the Jacobians of the curves $V$ (see series of works by V. M. Buchstaber, V. Z. Enolskii and D. V. Leikin).
In our recent works with A. V. Mikhailov a method of constructing polynomial dynamical systems on the basis of $k$-th symmetric powers of the curve $V$, where $k$ is not obligatory equal to $g$, has been developed. For $k=2$ an explicit realization of this method has been given and a Lie algebra of differentiation of solutions over parameters of the curve has been described.
This method has led to the problem: find solutions of the dynamical systems constructed in functions in $C^g$, such that for $k<g$ they are not $2g$-periodic, but the restriction of this functions to the image of Abel–Jacobi map for the $k$-th symmetric power of the genus $g$ curve is $2g$-periodic.
In the talk in an explicit form integrable polynomial dynamical systems in $C^4$ will be presented. They are constructed on the basis of symmetric squares of hyperelliptic curves of genus $g>0$. For $g=1$ a solution in terms of classic Weierstrass functions will be described. For $g=2$ a solution in two-dimensional analogues of Weierstrass functions will be described and the Lie algebra of differentiation of this functions over parameters of the curve obtained in a recent work of the author will be presented. For $g=3$ a solution in meromorphic functions on the sigma-divisor of hyperelliptic genus 3 curve obtained in a recent work of the author and Takanori Ayano will be described.