Infinite-Dimensional Lie Algebras Determined by the Space of Symmetric Squares of Hyperelliptic Curves

We construct Lie algebras of vector fields on universal bundles of symmetric squares of hyperelliptic curves of genus $g=1,2,\dots$. For each of these Lie algebras, the Lie subalgebra of vertical fields has commuting generators, while the generators of the Lie subalgebra of projectable fields determines the canonical representation of the Lie subalgebra with generators $L_{2q}$, $q=-1, 0, 1, 2, \dots$, of the Witt algebra. As an application, we obtain integrable polynomial dynamical systems.