1:−31:−3 resonant centers on C2C2 with homogeneous cubic nonlinearities

We characterize the collection of systems of differential equations on C2C2 of the form ẋ=x+p(x,y), ẏ=−3y+q(x,y), where pp and qq are homogeneous polynomials of degree three (either of which may be zero), that possess a first integral in a neighborhood of (0,0)(0,0) of the form x3y+⋯x3y+⋯, where omitted terms are of order at least five. Such systems are called 1:−31:−3 resonant centers.