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Existence Conditions for Darboux Curves and Analytic First Integrals of a Liénard-Type Quadratic Vector Field

  • Shaoxuan Huang
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Abstract

We study the rational quadratic differential equation dy/dx = (ay2+by+cx)/y2, a,b,c∈C, under the non-degeneracy assumptions c≠ 0, 2ay+bnot≡ 0. Equivalently, after clearing the denominator, we consider the polynomial vector field ẋ=y2, ẏ=ay2+by+cx. We give a complete, directly checkable classification of its Darboux curves. If a=0, no non-constant Darboux polynomial exists. If a≠ 0, a non-constant Darboux polynomial exists if and only if c=−ab or c=−2ab. In these two cases the unique irreducible Darboux polynomials, up to non-zero constant multiples, are respectively y−ax, y2−2bx. 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.

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Cite

BibTeXRIS
BibTeX
@misc{huang2026existence,
  title = {{Existence Conditions for Darboux Curves and Analytic First Integrals of a Liénard-Type Quadratic Vector Field}},
  author = {Shaoxuan Huang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.13325},
  eprint = {2609.13325},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.13325},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - UNPB
TI  - Existence Conditions for Darboux Curves and Analytic First Integrals of a Liénard-Type Quadratic Vector Field
AU  - Shaoxuan Huang
PY  - 2026
DA  - 2026/10/10
PB  - arXiv
DO  - 10.48550/arxiv.2609.13325
UR  - https://arxiv.org/abs/2609.13325
AB  - We study the rational quadratic differential equation dy/dx = (ay^2+by+cx)/y^2, a,b,c∈C, under the non-degeneracy assumptions c≠ 0, 2ay+bnot≡ 0. Equivalently, after clearing the denominator, we consider the polynomial vector field ẋ=y^2, ẏ=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≠ 0, a non-constant Darboux polynomial exists if and only if c=−ab or c=−2ab. In these two cases the unique irreducible Darboux polynomials, up to non-zero constant multiples, are respectively y−ax, 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.
ER  -