{
  "abstract": "We study the rational quadratic differential equation \\[ \\frac{\\mathrm{d}y}{\\mathrm{d}x} = \\frac{ay^2+by+cx}{y^2}, \\qquad a,b,c\\in\\C, \\] under the non-degeneracy assumptions \\[ c\\neq 0,\\qquad 2ay+b\\not\\equiv 0. \\] Equivalently, after clearing the denominator, we consider the polynomial vector field \\[ \\dot{x}=y^2,\\qquad \\dot{y}=ay^2+by+cx. \\] We give a complete, directly checkable classification of its Darboux curves. If $a=0$, no non-constant Darboux polynomial exists. If $a\\neq 0$, a non-constant Darboux polynomial exists if and only if \\[ c=-ab\\qquad\\text{or}\\qquad c=-2ab. \\] In these two cases the unique irreducible Darboux polynomials, up to non-zero constant multiples, are respectively \\[ y-ax,\\qquad y^2-2bx. \\] Consequently every non-constant Darboux polynomial is a non-zero constant multiple of a positive integral power of the corresponding irreducible factor. We then place this classification in the framework of Riccati (R-)integrability and rational potentials. Excluding the exceptional branch $c=-2ab$, the analytic integrability theorem identifies $c=-ab$ as the branch admitting a global Riccati-type analytic first integral. For $c=-2ab$, we compute the first four transverse variational groups along the transformed Darboux divisor over the rational function field. Their dimensions are $1,2,3,4$; the fourth is the full group of invertible fourth-order transverse jets. We prove that this exceptional branch admits no Riccati first integral and hence is not R-integrable. Finally, we relate the branch $c=-ab$ to the Quartic Inverse Riccati (QIR) class and discuss an invariant-based classification problem for quartic Abel equations.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.13325",
  "author_total": 1,
  "authors": [
    {
      "name": "Shaoxuan Huang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ewz5t8dcp8v2w.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13325",
  "fulltext_url": null,
  "id": "ewz5t8dcp8v2w",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13325",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.SC"
    ],
    [
      "arxiv",
      "nlin.SI"
    ],
    [
      "openalex",
      "2608|Advanced Differential Equations and Dynamical Systems"
    ]
  ],
  "openalex": "W7213345133",
  "pdf_url": "https://arxiv.org/pdf/2609.13325",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ewz5t8dcp8v2w.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:06Z",
      "native_id": "2609.13325",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13325"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:nlin.SI",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Existence Conditions for Darboux Curves and Analytic First Integrals of a Liénard-Type Quadratic Vector Field",
  "title_text": "Existence Conditions for Darboux Curves and Analytic First Integrals of a Liénard-Type Quadratic Vector Field",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:06Z",
  "url": "https://research.draconitic.com/paper/ewz5t8dcp8v2w",
  "venue": "arXiv",
  "versions": []
}