Abstract
The generalized Liénard equation, an important subclass of polynomial planar vector fields closely related to the limit-cycle problem in the second part of Hilbert’s 16th problem, has been extensively studied for real limit cycles, whose number depends sensitively on the degrees of the involved polynomials. This model also finds wide applications in electrical engineering, mathematical biology, population dynamics, nonlinear chemistry, seismology and economic fluctuation research. In this work, we systematically investigate meromorphic solutions of the generalized Liénard system. Using Nevanlinna theory together with the Kowalevski–Gambier method, we carry out a rigorous full classification of all admissible meromorphic solutions and prove that every non-constant meromorphic solution must be an elliptic function or one of its degenerate forms. This study extends the classical real-domain qualitative theory of Liénard systems to the complex plane and provides new complex-analytic perspectives for exploring the global structure of nonlinear oscillations.