Centers for the Kukles homogeneous systems with odd degree

Abstract

For the polynomial differential system , , where is a homogeneous polynomial of degree there are the following two conjectures raised in 1999. (1) Is it true that the previous system for has a center at the origin if and only if its vector field is symmetric about one of the coordinate axes? (2) Is it true that the origin is an isochronous center of the previous system with the exception of the linear center only if the system has even degree? We prove both conjectures for all odd.