1:−3 resonant centers on with homogeneous cubic nonlinearities

We characterize the collection of systems of differential equations on of the form , , where p and q are homogeneous polynomials of degree three (either of which may be zero), that possess a first integral in a neighborhood of (0,0) of the form x3y+ , where omitted terms are of order at least five. Such systems are called 1:−3 resonant centers.