Finding solutions of parabolic Monge–Ampère equations by using the geometry of sections of the contact distribution

In a series of papers we have described normal forms of parabolic Monge–Ampère equations (PMAEs) by means of their characteristic distribution. In particular, PMAEs with two independent variables are associated with Lagrangian (or Legendrian) subdistributions of the contact distribution of a 5-dimensional contact manifold. The geometry of sections of the contact distribution allowed us to get the aforementioned normal forms. In the present work, for a distinguished class of PMAEs, we will construct 3-parametric families of solutions starting from particular sections of the characteristic distribution. We will illustrate the method by several concrete computations. Moreover, we will see, for some linear PMAEs, how to construct a recursive process for obtaining new solutions. At the end, after showing that some classical equations on affine connected 3-dimensional manifolds are PMAEs, we will apply the integration method to some particular examples.