{
  "abstract": "We study the two fourth-order quantum superintegrable systems in the Euclidean plane that separate in polar coordinates and whose angular potentials are determined by the general four-parameter sixth Painlevé transcendent. One system has an oscillator radial term and the other an extended-Coulomb radial term. For both systems we give the fourth-order quantum integral explicitly, derive the cubic symmetry algebra and its Casimir, and construct a deformed-oscillator realization. At fixed energy the two algebras are specializations of the Heun--Racah algebra. Their structure functions contain the same quartic angular factor, determined by the Painlevé~VI parameters, while different radial factors distinguish oscillator and Coulomb dynamics. The Coulomb integral is obtained by transforming the Cartesian Hermitian symmetry at the operator level. The lower-order terms generated by the polar coordinate change are essential for commutation with the Hamiltonian. The two quantum systems are also related by the Levi-Civita transformation and coupling-constant metamorphosis, which gives a direct correspondence between their Painlevé parameters, algebras, Casimirs and representations. Earlier exceptional-Jacobi models arise at special rational parameter values, whereas the results here apply to the full four-parameter Painlevé~VI family. We also compare the quantum results with the corresponding classical Poisson algebras.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.00322",
  "author_total": 2,
  "authors": [
    {
      "name": "Ian Marquette"
    },
    {
      "name": "Sarah Post"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/eyjqq35ez4psn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.00322",
  "fulltext_url": null,
  "id": "eyjqq35ez4psn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.00322",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.CA"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "nlin.SI"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.00322",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/eyjqq35ez4psn.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:20:59Z",
      "native_id": "2611.00322",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.00322"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:nlin.SI",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Painlevé VI superintegrability and Heun-Racah algebras",
  "title_text": "Painlevé VI superintegrability and Heun-Racah algebras",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/eyjqq35ez4psn",
  "venue": "arXiv",
  "versions": []
}