Internal, external, and generalized symmetries

Bäcklund′s theorem, which characterizes contact transformations, is generalized to give an analogous characterization of "internal symmetries" of systems of differential equations. For a wide class of systems of differential equations, we prove that every internal symmetry comes from a first order generalized symmetry and, conversely, every first order generalized symmetry satisfying certain explicit contact conditions determines an internal symmetry. We analyze the contact conditions in detail, deducing powerful necessary conditions for a system of differential equations that admit "genuine" internal symmetries, i.e., ones which do not come from classical "external" symmetries. Applications include a direct proof that both the internal symmetry group and the first order generalized symmetries of a remarkable differential equation due to Hilbert and Cartan are the noncompact real form of the exceptional simple Lie group G₂.