Hyperelliptic sigma functions and the sequence of the Novikov's $g$-equations

In 1974, S. P. Novikov discovered an algebro-geometrical method for constructing periodic and quasi-periodic solutions of the KdV equation. He introduced the g-stationary equations of the KdV-hierarchy (namely the Novikov's $g$-equations) which correspond to integrable polynomial dynamical systems in $\mathbb{C}^{3g}$ with $2g$ polynomial integrals.
The talk is devoted to differential equations and dynamical systems, which are integrable in hyperelliptic sigma functions.
We will introduce systems of $2g$-dimensional heat equations in a nonholonomic frame which define this functions. The operators of such system generate a polynomial Lie algebra with only three generators for $g > 1$. We will construct an infinite-dimensional polynomial dynamical system that is universal for all polynomial dynamical systems corresponding to the sequence of Novikov's $g$-equations.