An introduction to completely exceptional 2nd order scalar PDEs

In his 1954 paper about the initial value problem for 2D hyperbolic nonlinear PDEs, P. Lax declared that he had “a strong reason to believe” that there must exist a well-defined class of “not genuinely nonlinear” nonlinear PDEs. In 1978 G. Boillat coined the term “completely exceptional” to denote it. In the case of second order (nonlinear) PDEs, he also proved that this class reduces to the class of Monge–Ampère equations. We review here, against a unified geometric background, the notion of complete exceptionality, the definition of a Monge–Ampère equation, and the interesting link between them.