An inverse approach to the center-focus problem for polynomial differential system with homogenous nonlinearities

We study the particular case of centers which have a local analytic first integral of the form H=12(x2+y2)(1+∑j=1∞ϒj), in a neighborhood of the origin, where ϒj is a convenient homogenous polynomial of degree j, for j≥1. These centers are called weak centers, they contain the class of center studied by Alwash and Lloyd, the uniform isochronous centers and the isochronous holomorphic centers, but they do not coincide with the class of isochronous centers. We give a classification of the weak centers for quadratic and cubic vector fields with homogenous nonlinearities.