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The set $\mathbf{F}(3)$ of foliations of degree three on the complexprojective plane can be identified with a Zariski's open set of a projectivespace 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 flatLegendre transform (dual web) is a Zariski closed subset of $\mathbf{F}(3)$. Weclassify 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 thisclassification we deduce that $\mathbf{FP}(3)$ has exactly $12$ irreduciblecomponents. We also deduce that up to automorphism there are $4$ convexfoliations of degree three on $\mathbb{P}^{2}_{\mathbb{C}}.$

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