Integrable systems and symmetric products of curves

We show how there is associated to each non-constant polynomial $F(x,y)$ a completely integrable system with polynomial invariants on $\Rd$ and on $\C{2d}$ for each $d\geq1$; in fact the invariants are not only in involution for one Poisson bracket, but for a large class of polynomial Poisson brackets, indexed by the family of polynomials in two variables. We show that the complex invariant manifolds are isomorphic to affine parts of $d$-fold symmetric products of a deformation of the algebraic curve $F(x,y)=0$, and derive the structure of the real invariant manifolds from it. We also exhibit Lax equations for the hyperelliptic case (i.e., when $F(x,y)$ is of the form $y^2+f(x)$) and we show that in this case the invariant manifolds are affine parts of distinguished (non-linear) subvarieties of the Jacobians of the curves. As an application the geometry of the H\'enon-Heiles hierarchy --- a family of superimposable integrable polynomial potentials on the plane --- is revealed and Lax equations for the hierarchy are given.