{
  "abstract": "The $S$-function method for rational second order ODEs $y''=φ(x,y,y')$ uses $S=I_y/I_z$, attached to a first integral $I$, which satisfies a Riccati equation along the flow; it was designed for Liouvillian $I$. We study the case of rational $S$ with non-Liouvillian $I$. From any rational $g(x,y,z)$ with $g_z\\neq0$ and any linear equation $A_2w''+A_1w'+A_0w=0$ with polynomial coefficients, the substitution $w'/w=g(x,y,y')$ yields a rational 2ODE whose first integral is built from two independent solutions, and $S=g_y/g_z$: the Wronskian cancels, so $S$ does not see the linear equation, and one rational $S$ is shared by equations with and without Liouvillian first integrals. Conversely, given $(φ,S)$, membership in this class reduces to a rational first integral of an explicit planar vector field plus a Möbius-invariant Riccati test in $x$, which recovers the linear equation. A transfer theorem shows that $I$ is Liouvillian if and only if this equation has a Liouvillian solution, so Liouvillianity is decidable on the class by Kovacic's algorithm: a three-variable problem with no a priori degree bound is reduced to two classical ones. On nine pull-back equations of degree up to $15$, given without information on their construction, the procedure recovers the Airy equation and the Bessel equation of order zero.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.32769",
  "author_total": 2,
  "authors": [
    {
      "name": "L. G. S. Duarte"
    },
    {
      "name": "L. A. C. P. da Mota"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1x3tf4fhakrxw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.32769",
  "fulltext_url": null,
  "id": "1x3tf4fhakrxw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.32769",
  "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"
    ],
    [
      "openalex",
      "2608|Advanced Differential Equations and Dynamical Systems"
    ]
  ],
  "openalex": "W7223702394",
  "pdf_url": "https://arxiv.org/pdf/2610.32769",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1x3tf4fhakrxw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:15Z",
      "native_id": "2610.32769",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.32769"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Rational $S$-functions for second order ODEs with non-Liouvillian first integrals",
  "title_text": "Rational S-functions for second order ODEs with non-Liouvillian first integrals",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/1x3tf4fhakrxw",
  "venue": "arXiv",
  "versions": []
}