The cubic Darboux systems with a non-elementary singular point at the Poincaré equator

We study the global behavior of the trajectories of the polynomial system $\dot{x}=x-x^{2}y+px{y}^2+y^3,\dot{y}=y+p{y}^3,p\in \mathbb{R}$. Our study is related to the paper {\it Alarcon B., Castro S.B.S.D., Labouriau I.S.} Glodal planar dynamics with star nodes: beyond Hilbert’s 16th problem//arXiv:2106/07516v2 [math.DS].