  This paper studies global webs on the projective plane with vanishing
curvature. The study is based on an interplay of local and global arguments.
The main local ingredient is a criterium for the regularity of the curvature at
the neighborhood of a generic point of the discriminant. The main global
ingredient, the Legendre transform, is an avatar of classical projective
duality in the realm of differential equations. We show that the Legendre
transform of what we call reduced convex foliations are webs with zero
curvature, and we exhibit a countable infinity family of convex foliations
which give rise to a family of webs with zero curvature not admitting
non-trivial deformations with zero curvature.
