Quartic rigid systems in the plane and in the Poincaré sphere

We consider the planar family of rigid systems of the form $x^{\mathrm{\prime }}=-y+xP(x,y),y^{\mathrm{\prime }}=x+yP(x,y)$, where $P$ is any polynomial with monomials of degree one and three. This is the simplest non-trivial family of rigid systems with no rotatory parameters. The family can be compactified to the Poincaré sphere such that the vector field along the equator is not identically null. We study the centers, singular points and limit cycles of that family on the plane and on the sphere.