Center conditions for a class of planar rigid polynomial di-erential systems

In general the center-focus problem cannot be solved, but in the case that the singularity has purely imaginary eigenvalues there are algorithms to solving it. The present paper implements one of these algorithms for the polynomial di-erential systems of the form x = -y + x∫(x)g(y); y = x + y∫(x)g(y); where f(x) and g(y) are arbitrary polynomials. These di-erential systems have constant angular speed and are also called rigid systems. More precisely, in this paper we give the center conditions for these systems, i.e. the necessary and su-cient conditions in order that they have an uniform isochronous center. In particular, the existence of a focus with the highest order is also studied.