On the saddle order of polynomial differential systems at a resonant singular point

In this paper, we study the complexity of integrability of planar polynomial differential systems whose eigenvalues admit resonances at a saddle singular point. We prove that for arbitrary integer n≥2, if one of n+2 and 2n+1 is a prime number, then there exists a polynomial differential system of degree n with 1:−2 resonance at its saddle singular point such that the saddle order can be as high as n2−1.