Isochronicity and Commutation of Polynomial Vector Fields

Abstract

We study a connection between the isochronicity of a center of a polynomial vector field and the existence of a polynomial commuting system. We demonstrate an isochronous system of degree 4 which does not commute with any polynomial system. We prove that among the Newton polynomial systems only the Lienard and Abel systems may commute with transversal polynomial fields. We give a full and constructive description of centralizers of the Abel polynomial systems. We give new examples of isochronous systems.