{
  "annotation_id": "77cbdd12-d05a-48af-87fc-f52c86707100",
  "date_created": "2026-03-02T18:02:48.244000Z",
  "date_modified": "2026-03-02T18:02:48.244000Z",
  "file_hash": "d8c99a5d90a5b2cc902a2778eadd73de5bc7cc5b9061700b30640455431ee6f3",
  "private": false,
  "record": {
    "abstract": "We show how there is associated to each non-constant polynomial $F(x,y)$ a\ncompletely integrable system with polynomial invariants on $\\Rd$ and on\n$\\C{2d}$ for each $d\\geq1$; in fact the invariants are not only in involution\nfor one Poisson bracket, but for a large class of polynomial Poisson brackets,\nindexed by the family of polynomials in two variables. We show that the complex\ninvariant manifolds are isomorphic to affine parts of $d$-fold symmetric\nproducts of a deformation of the algebraic curve $F(x,y)=0$, and derive the\nstructure of the real invariant manifolds from it. We also exhibit Lax\nequations for the hyperelliptic case (i.e., when $F(x,y)$ is of the form\n$y^2+f(x)$) and we show that in this case the invariant manifolds are affine\nparts of distinguished (non-linear) subvarieties of the Jacobians of the\ncurves. As an application the geometry of the H\\\u0027enon-Heiles hierarchy --- a\nfamily of superimposable integrable polynomial potentials on the plane --- is\nrevealed and Lax equations for the hierarchy are given.",
    "arxiv_id": "solv-int/9402002",
    "authors": [
      "Pol Vanhaecke"
    ],
    "categories": [
      "solv-int",
      "nlin.SI"
    ],
    "title": "Integrable systems and symmetric products of curves",
    "url": "https://arxiv.org/abs/solv-int/9402002"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "0a939ad6-5387-442d-bbc8-f39b6227fec7",
    "id": "arXiv Dataset IDs",
    "type": "Model",
    "variant": "snapshot-2026-03-01",
    "version": "0.1.0"
  },
  "user_id": 1000002
}