发表机构
Chinese univeristy of Hongkong, ShenZhen（香港中文大学（深圳））
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文完整分类了Liénard型二次向量场的Darboux曲线，给出存在条件（c=-ab或c=-2ab），并证明全局Riccati型解析首次积分仅在c=-ab分支出现，且与QIR类相关。
AI 中文摘要
我们研究有理二次微分方程 \\[ \frac{\mathrm{d}y}{\mathrm{d}x} = \frac{ay^2+by+cx}{y^2}, \qquad a,b,c\in\C, \\] 在非退化假设 \\[ c\neq 0,\qquad 2ay+b\not\equiv 0 \\] 之下。等价地，在清除分母后，我们考虑多项式向量场 \\[ \dot{x}=y^2,\qquad \dot{y}=ay^2+by+cx. \\] 我们给出其Darboux曲线的完整且可直接检验的分类。若$a=0$，则不存在非常数Darboux多项式。若$a\neq 0$，则存在非常数Darboux多项式当且仅当 \\[ c=-ab\qquad\text{或}\qquad c=-2ab. \\] 在这两种情形下，唯一的不可约Darboux多项式（在非零常数倍意义下）分别为 \\[ y-ax,\qquad y^2-2bx. \\] 因此，每个非常数Darboux多项式都是对应不可约因子的正整数幂的非零常数倍。随后我们将此分类置于Riccati（R-）可积性与有理势的框架中，并阐述解析可积性定理，断言全局Riccati型解析首次积分恰好出现在分支$c=-ab$上。最后，我们将该分支与四次逆Riccati（QIR）类联系起来，并讨论四次Abel方程基于不变量的分类问题。
英文摘要
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.
