Integrability of centers perturbed by quasi-homogeneous polynomials
A polynomial \(X(x,y)\) is called \((p-q)\) quasi-homogeneous of degree \(r\) if the identity \[ X(c^px,c^qy)=c^rX(x,y) \] holds for any real \(c\). The authors consider two-dimensional autonomous systems of the form \[ x'=-y^{2p-1}+X(x,y),\qquad y'=x^{2q-1}+Y(x,y), \] where \(X\) and \(Y\) are \((p-q)\) quasi-homogeneous polynomials of degree \(2pq-q+1\) and \(2pq-p+1\), respectively. Necessary and sufficient conditions for the integrability of this system are given in terms of coefficients of \(X\) and \(Y\).
scientific article; zbMATH DE number 1031648