Consider the planar system (1) \(\dot x=F(x)\), \(x\in\mathbb{R}^2\), \(f\in C^1(\mathbb{R}^2,\mathbb{R}^2)\), let \(O\) be a singular point of system (1), and let \(\lambda_1(x)\), \(\lambda_2(x)\) denote the eigenvalues of the Jacobian matrix of \(f\). \textit{L. Markus} and \textit{H. Yamabe} [Osaka Math. J. (2) 12, 305--317 (1960; Zbl 0096.28802)] conjectured that if \(\lambda_1(x)\) and \(\lambda_2(x)\) have negative real parts for all \(x\in\mathbb{R}^2\), then \(O\) is globally asymptotically stable. They were able to show that the conjecture is true for systems which are equivalent to second order differential equations. \textit{G. Meisters} and \textit{C. Olech} [Analyse mathématique et applications, Contrib. Honneur Jacques-Louis Lions, 373--381 (1988; Zbl 0668.34048)] proved the conjecture to be true for systems where the components of \(f\) are polynomials. The author extends their proof to systems for which only one of the components of \(f\) is a polynomial or, more generally, a function whose level sets have less than \(N\) connected components, for a fixed integer \(N\). As a consequence, he also obtains a different proof of the result of Markus and Yamabe.