Liouvillian Integrability Versus Darboux Polynomials

In this note we provide a sufficient condition on the existence of Darboux polynomials of polynomial differential systems via existence of Jacobian multiplier or of Liouvillian first integral and a degree condition among different components of the system. As an application of our main results we prove that the Liénard polynomial differential system x˙=y,y˙=-f(x)y-g(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{x}=y,\, \dot{y}=-f(x)y-g(x)$$\end{document} with degf>degg\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\deg f>\deg g$$\end{document} is not Liouvillian integrable.