For the polynomial differential system x˙ = −y, y˙ = x + Qn(x, y), where Qn(x, y) is a homogeneous polynomial of degree n there are the following two conjectures raised in 1999. (1) Is it true that the previous system for n >= 2 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 isochrono…