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We construct Lie algebras of vector fields on universal bundles$\mathcal{E}^2_{N,0}$ of symmetric squares of hyperelliptic curves of genus$g=1,2,\dots$, where $g=\left[\frac{N-1}{2}\right], \ N=3,4,\ldots$. For eachof these Lie algebras, the Lie subalgebra of vertical fields has commutinggenerators, while the generators of the Lie subalgebra of projectable fieldsdetermines the canonical representation of the Lie subalgebra with generators$L_{2q}$, $q=-1, 0, 1, 2, \dots$, of the Witt algebra. We give explicitly abi-rational equivalence of the space $\mathcal{E}^2_{N,0}$ and $\bbbC^{N+1}$(in the case $N=5$ it is a well known result of Dubrovin and Novikov) andconstruct a polynomial Lie algebra on $\bbbC^{N+1}$, which contains twocommuting generators. These commuting generators results in two compatiblepolynomial dynamical systems on $\bbbR^4$, which possess two common polynomialfirst integrals. Moreover, these systems are Hamiltonian and thus Liouvilleintegrable. Using Abel-Jacobi two point map the solutions of these systems canbe given in terms of functions defined on universal covering of the universalbundle of the Jacobians of the curves. These functions are not Abelian if $g\ne2$. Finally we give explicit solutions of the constructed Hamiltonian systemson $\bbbR^4$ in the cases $N=3,4,5$.

The space of symmetric squares of hyperelliptic curves and integrable Hamiltonian polynomial systems on $\bbbR^4$