**Abstract: **We have found a quite general construction of commuting vector fields on the $k$-th symmetric power of $\mathbb{C}^{m}$ and tangent vector fields to the $k$-th symmetric power of an affine variety $V\subset\mathbb{C}^{m}$. Application of this construction to the $k$-th symmetric power of a plane algebraic curve $V_g$ of genus $g$ leads to $k$ integrable Hamiltonian systems on $\mathbb{C}^{2k}$ (or on $\mathbb{R}^{2k}$, if the base field is $\mathbb{R}$). In the case $k=g$, the symmetric power ${\rm Sym}^k(V_g)$ is birationally isomorphic to the Jacobian of the curve $V_g$, and our system is equivalent to well-known Dubrovin's system, which was derived and studied in the theory of finite-gap solutions (algebro-geometric integration) of the Korteweg–de Vries equation. We have found coordinates in which the obtained systems and their Hamiltonians are polynomial. For $k=2,\ g=1,2,3$ we present these systems explicitly and discuss the problem of their integration.

**Venue: **7th corpus YarSU, lecture theatre 419