On integrability of differential equations defined by the sum of homogeneous vector fields with degenerate infinity

The paper deals with polynomials systems with degenerate infinity from diff erent points of view. We show the utility of the projective techniques for such systems, and a more detailed study in the quadratic and cubic cases is carried out. On the other hand, some results on Darboux integrability in t he affine plane for a class of systems are given. In short we show the expl icit form of generalized Darboux inverse integrating factors for the above kind of systems. Finally, a short proof of the center cases for arbitrary d egree homogeneous systems with degenerate infinity is given, and moreover w e solve the center problem for quartic systems with degenerate infinity and constant angular speed.