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

We study the rational quadratic differential equation $$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{a{y}^2+by+cx}{y^2},a,b,c\in \text{C},$$ under the non-degeneracy assumptions $$c\mathrm{\ne }0,2ay+b\equiv ̸0.$$ Equivalently, after clearing the denominator, we consider the polynomial vector field $$\dot{x}=y^2,\dot{y}=a{y}^2+by+cx\mathrm{.}$$ We give a complete, directly checkable classification of its Darboux curves. If $a=0$, no non-constant Darboux polynomial exists. If $a\mathrm{\ne }0$, a non-constant Darboux polynomial exists if and only if $$c=-ab\text{or}c=-2ab\mathrm{.}$$ In these two cases the unique irreducible Darboux polynomials, up to non-zero constant multiples, are respectively $$y-ax,y^2-2bx\mathrm{.}$$ 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, and formulate the analytic integrability theorem asserting that a global Riccati-type analytic first integral occurs precisely on the branch $c=-ab$. Finally, we relate this branch to the Quartic Inverse Riccati (QIR) class and discuss an invariant-based classification problem for quartic Abel equations.