Geometry of certain foliations on the complex projective plane

Let $d\geq2$ be an integer. The set $\mathbf{F}(d)$ of foliations degree $d$ on the complex projective plane can identified with a Zariski's open space dimension $d^2+4d+2$ which $\mathrm{Aut}(\mathbb P^2_{\mathbb C})$ acts. We show that there are exactly two orbits $\mathcal{O}(\mathcal{F}_{1}^{d})$ and $\mathcal{O}(\mathcal{F}_{2}^{d})$ minimal $6$, necessarily closed in $\mathbf{F}(d)$. This generalizes known results degrees $2$ $3.$ deduce orbit $\mathcal{O}(\mathcal{F})$ element $\mathcal{F}\in\mathbf{F}(d)$ $7$ is if only $\mathcal{F}_{i}^{d}\not\in\overline{\mathcal{O}(\mathcal{F})}$ for $i=1,2.$ allows us to any $d\geq3$ $\mathbf F(d)$ other than $\mathcal{O}(\mathcal{F}_{2}^{d}),$ unlike situation $2.$ On hand, we introduce notion basin attraction $\mathbf{B}(\mathcal{F})$ foliation as $\mathcal{G}\in\mathbf{F}(d)$ such $\mathcal{F}\in\overline{\mathcal{O}(\mathcal{G})}.$ $\mathbf{B}(\mathcal{F}_{1}^{d})$, resp. $\mathbf{B}(\mathcal{F}_{2}^{d})$, contains quasi-projective subvariety greater or equal $\dim\mathbf{F}(d)-(d-1)$, $\dim \mathbf{F}(d)-(d-3)$. In particular, obtain $\mathbf{B}(\mathcal{F}_{2}^{3})$ non-empty Zariski subset $\mathbf{F}(3)$. analog $3$ result due Cerveau, D\'eserti, Garba Belko Meziani.