The aim of my talk is to describe another approach, based on the ideas from geometric control theory, which allows to overcome the difficulties mentioned above. This approach allows to reduce the equivalence problem for vector distributions to the study of curves of flags of isotropic/coisotropic subspaces in a sympectic space. Our classification of distributions is done according to a so-called Young diagram of these curves of flags and is not directly related to Tanaka symbols of the distribution itself. The local geometry of distributions can be recovered from the properties of symmetry groups of so-called flat curves of flags associated with its Young diagram. For any given Young diagram one can describe the flat distribution and construct a canonical frame for any other distribution (with the same Young diagram). In the case of rank 3 distributions with non-rectangular Young diagram the infinitesimal symmetry algebra of the flat distribution can be described in terms of rational normal curves (their secants and tangential developable) in projective spaces.

The classification of (in general nonlinear) scalar 2nd order PDE in the plane into elliptic, parabolic, hyperbolic classes is a contact-invariant classification. Moreover, in the hyperbolic case, a finer contact-invariant classification reveals three subclasses: equations of Monge--Ampere, Goursat, and generic type. An intriguing property about hyperbolic equations of generic type is that any equation in this class admits at most a nine-dimensional (contact) symmetry group. This is in stark contrast to the Monge--Ampere class which contains the wave equation, admitting an infinite-dimensional symmetry group. In this talk, I will describe some of the basic contact invariants that arise in the theory and give an outline of how the nine-dimensional bound is established using further tools from exterior differential systems and Cartan's method of equivalence. The nine-dimensional bound is sharp: I'll also describe how normal forms for the contact-equivalence classes of maximally symmetric generic hyperbolic equations were found as well as the symmetry algebras which arise.