1:−3 resonant centers on C2 with homogeneous cubic nonlinearities

Abstract

We characterize the collection of systems of differential equations on C 2 of the form x ̇ = x + p ( x , y ) , y ̇ = − 3 y + q ( x , y ) , 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 x 3 y + ⋯ , where omitted terms are of order at least five. Such systems are called 1 : − 3 resonant centers.