Isochronous centers of a linear center perturbed by fifth degree homogeneous polynomials

In this work we study isochronous centers of two-dimensional autonomous system in the plane with linear part of center type and nonlinear part given by homogeneous polynomials of fifth degree. A complete classification of the necessary conditions for the time-reversible systems of this class is given in order to have an isochronous center at the origin. An open problem is stated for the sufficient conditions. Moreover, we find two nonreversible isochronous families from the center cases known. All the computations in order to obtain necessary conditions for such isochronous centers are given in polar coordinates and we give a proof of the isochronicity of these systems by using different methods.