Classification of foliations of degree three on mathbb{P}²_{mathbb{C}} with a flat Legendre transform

The set $\mathbf{F}(3)$ of foliations of degree three on the complex projective plane can be identified with a Zariski's open set of a projective space of dimension $23$ on which acts $\mathrm{Aut}(\mathbb{P}^{2}_{\mathbb{C}})$. The subset $\mathbf{FP}(3)$ of $\mathbf{F}(3)$ consisting of foliations of $\mathbf{F}(3)$ with a flat Legendre transform (dual web) is a Zariski closed subset of $\mathbf{F}(3)$. We classify up to automorphism of $\mathbb{P}^{2}_{\mathbb{C}}$ the elements of $\mathbf{FP}(3)$. More precisely, we show that up to automorphism there are $16$ foliations of degree three with a flat Legendre transform. From this classification we deduce that $\mathbf{FP}(3)$ has exactly $12$ irreducible components. We also deduce that up to automorphism there are $4$ convex foliations of degree three on $\mathbb{P}^{2}_{\mathbb{C}}.$