Integrability conditions of a weak saddle in generalized Liénard-like complex polynomial differential systems

We consider the complex differential system \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathop x\limits^. \, = \,x\, + \,yf(x),\,\,\,\,\,\,\,\mathop y\limits^. \, = \, - y\, + \,xf(y),$$\end{document}