Blow-up method to compute necessary conditions of integrability for planar differential systems

摘要

We provide a method to compute the necessary conditions of integrability for planar differential systems with a weak saddle at the origin. The method is used to characterize the local analytic integrability for resonant saddles of complex systems of the form x˙=px, and y˙=−y+f(y)x, 0≠p∈N, with f analytic at the origin and with f(0)=0.