Contact linearization of Monge-Ampére equations and Laplace invariants

Abstract

. A distinguishing feature of suchequations in the class of all second-order equations isthat it is closed with respect to contact transformations.This fact was known even to Sophus Lie, who studiedMonge–Ampere equations by methods of contactgeometry, which he created. In the 1870s and the1880s, Lie posed the problem of classifying theMonge–Ampere equations with respect to the(pseudo)group of contact transformations and, in par-ticular, reducing Monge–Ampere equations to quasilin-ear form (corresponding to D = 0 in Eq. (1)) and findingthe simplest coordinate representation of such equa-tions [8, 9].He also considered classification problems forMonge–Ampere equations admitting intermediate inte-grals. Lie himself found conditions for reducingMonge–Ampere equations to quasilinear form and tolinear equations with constant coefficients, but proofsof these theorems have never been published. Note,however, that verifying the presence of intermediateintegrals for general Monge–Ampere equations, muchmore constructing them, is a difficult problem.In 1983, V.V. Lychagin and V.N. Rubtsov consideredthe problem of classifying the Monge–Ampere equa-tions defined by (1) where the coefficients A, B, C, D,and E do not explicitly depend on the function v. They