Studies of the Petrov Module for a Family of Generalized Liénard Integrable Systems

Abstract

In this article we use the Lambert function in order to study a family of integrable generalized Lienard equations Xf which display a center. We first prove a conjugation lemma inside a continuum of nested periodic orbits. Then we deduce an explicit operator of Gelfand–Leray associated with the Hamiltonian of equation Xf. Afterwards, we provide a generating family for the associated Petrov module. Finally, by using the Lambert function, we study the monotonicity of the Abelian integral of this generating family’s elements.