Quartic Rigid Systems in the Plane and in the Poincaré Sphere

We consider the planar family of rigid systems of the form \(x'=-y+xP(x,y), y'=x+yP(x,y)\) $x^{\prime }=-y+xP(x,y),y^{\prime }=x+yP(x,y)$ , 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.