Normal forms for lagrangian distributions on 5-dimensional contact manifolds

A contact distribution CC on a manifold M   determines a symplectic bundle C→MC→M. In this paper we find normal forms for its lagrangian distributions by classifying vector fields lying in CC. Such vector fields are divided into three types and described in terms of the simplest ones (characteristic fields of 1st order PDE's). After having established the equivalence between parabolic Monge–Ampère equations (MAE's) and lagrangian distributions in terms of characteristics, as an application of our results we give normal forms for parabolic MAE's.