We identify a maximal system of equations which renders the classical Darboux problem elliptic, thereby providing a selection criterion for its well posedness. As a byproduct, we obtain, what we term symplectic factorization of vector fields: any map u, satisfying appropriate assumptions, can be factored as: u = χ◦ψ with ψ∗(ωm) = ωm,dχ[; ωm = 0 and ∇χ+(∇χ)t > 0. Here, ωm is the standard symplectic form on R2m,] is the musical isomorphism and [ its inverse. (This talk is based on a joint work with B. Dacorogna and O. Kneuss)