The topological classification of a family of quadratic differential systems in terms of affine invariant polynomials

In this paper we provide affine invariant necessary and sufficient conditions for a non-degenerate quadratic differential system to have an invariant conic $f(x, y)=0$ and a Darboux invariant of the form $f(x, y)^\lambda e^{st}$ with $\lambda,s\in \mathbb{R}$ and $s\ne0$. The family of all such systems has a total of seven topologically distinct phase portraits. For each one of these seven phase portraits we provide necessary and sufficient conditions in terms of affine invariant polynomials for a non-degenerate quadratic system in this family to possess this phase portrait.