  By studying the development of shock waves out of discontinuity waves, in
1954 P. Lax discovered a class of PDEs, which he called 'completely
exceptional', where such a transition does not occur after a finite time. A
straightforward integration of the completely exceptionality conditions allowed
Boillat to show that such PDEs are actually of Monge-Ampere type. In this
paper, we first recast these conditions in terms of characteristics, and then
we show that the completely exceptional PDEs, with 2 or 3 independent
variables, can be described in terms of the conformal geometry of the
Lagrangian Grassmannian, where they are naturally embedded. Moreover, for an
arbitrary number of independent variables, we show that the space of r-th
degree sections of the Lagrangian Grassmannian can be resolved via a BGG
operator. In the particular case of 1st degree sections, i.e., hyperplane
sections or, equivalently, Monge-Ampere equations, such operator is a close
analog of the trace-free second fundamental form.
