For a system in \(\mathbb {R}^n\) consisting of an unstable star node and homogeneous nonlinearity, i.e. \(\dot{x}=\mu x+Q(x)\) with \(\mu >0\) and Q(x) a homogeneous polynomial satisfying \(xQ(x)<0\) for all \(x\ne 0\) , Field in 1989 proved the existence of a unique topological invariant sphere. Here restricted to \(n=2\) , we prove that the topological invariant circle is of star shape, and obtain the classification of all their phase portraits in the Poincaré disc when Q is of degree 5 and the invariant circle is the circle \({\mathbb {S}}^1=\{(x,y,z) \in \mathbb {R}^3|\ x^2+y^2+z^2=1\}\) . Moreover, without the restriction \(xQ(x)<0\) we characterize the relation between equilibria (both finite and infinite) and the invariant straight lines passing through the origin. These results extend and improve the previous results of other authors.
