The space of symmetric squares of hyperelliptic curves: infnite-dimensional Lie algebras and polynomial integrable dynamical systems on $\mathbb{C}^4$

We construct Lie algebras of vector fields on universal bundles of symmetric squares of hyperelliptic curves of genus $g = 1, 2,\ldots$. For each of these Lie algebras, the Lie subalgebra of vertical fields has two commuting generators, while the generators of the Lie subalgebra of projectable vector fields determines the canonical representation of the Lie subalgebra with generators $L_{2q},\ q =-1, 0, 1, 2,\ldots$ , of the Witt algebra. The vertical vector fields yield two commuting integrable polynomial dynamical systems on $\mathbb{C}^4$, while the projectable fields provide us with the Lie algebra of derivations of the solutions with respect to the curve parameters. The method can be extended to higher symmetric powers and more general algebraic curves.
The talk is based on joint paper with V.M.Buchstaber