Flat 3-webs of degree one on the projective plane

The aim of this work is to study global $3$-webs with vanishing curvature. We wish to investigate degree $3$ foliations for which their dual web is flat. The main ingredient is the Legendre transform, which is an avatar of classical projective duality in the realm of differential equations. We find a characterization of degree $3$ foliations whose Legendre transform are webs with zero curvature.