Analytic integrability of certain resonant saddle

We provide sufficient conditions of analytic integrability for a family of planar differential system with a p : -q resonant saddle at the origin. The conditions are then shown to be necessary in several finite dimensional families where the calculations can be performed explicitly. It is conjectured that the condi-tions are in fact complete for all families. Though the form of the equations is quite simple, their study merits further attention as they exhibit an interesting dichotomy between finite and arbitrary dimensional components of the center variety. (c) 2021 Elsevier Ltd. All rights reserved.