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A. Gasull shared a list of 33 open problems in low dimensional dynamicalsystems in his work in 2021. The second part of Problem 3 is about whether thelimit cycle of a quasi-homogeneous system $  \dot{x}=y,\;  \dot{y}=-x^3+\alpha x^2y+y^3 $ is unique. In this paper, we give a positiveanswer to this question by analysing the uniqueness of the heteroclinicseparatrix at infinity.

A proof of the uniqueness of the limit cycle of a quasi-homogeneous system