Explicit expression for a first integral and phase portrait for a class of planar differential system

Abstract

where A1(x,y), A2(x,y), B1(x,y), B2(x,y), P(x,y), Q(x,y), R(x,y), S(x,y) are homogeneous polynomials of degree a, a, b, b, n, n, m, m respectively. Writing this system in polar coordinates, from which we can compute a formal First integral for the system. Finally, we give the curves which are formed by the trajectories for the system.