Integrability conditions of a weak saddle in a complex polynomial differential system (Q2070349)

The paper is concerned with the local analytic integrability of a complex differential system with a weak saddle \[ \dot{x}=x+y f(x), \quad \dot{y}=-y+x g(y), \] where \[ f(x)=\sum_{j=1}^{\infty} a_{j} x^{j}, \quad g(y)=\sum_{j=1}^{\infty} b_{j} y^{j} \] are analytic functions. When $a_1,b_1$ are not all zero, the integrability conditions are completely characterized. When $a_1=b_1=0$, the integrability conditions for fixed degree of $f$ and $g$ less than or equal to 6 are also given.