Isochronicity of a Z2-equivariant quintic system

The main purpose of this paper is to study the problem of simultaneous existence of two isochronous centers in some families of planar polynomial differential systems with symmetry. More precisely, we present conditions for a family of Z2-equivariant systems of degree 5 to have an isochronous bi-center. We also study their global phase portraits in the Poincaré disk and present considerations about the number of simultaneous centers in Z2-equivariant systems of degree 3 and 5. This study provides examples of quintic systems possessing three and five isochronous centers simultaneously.