Rank 2 distributions of Monge equations: Symmetries, equivalences, extensions

Abstract

By developing the Tanaka theory for rank 2 distributions, we completely classify classical Monge equations having maximal finite-dimensional symmetry algebras with fixed (albeit arbitrary) pair of its orders. Investigation of the corresponding Tanaka algebras leads to a new Lie-Bäcklund theorem. We prove that all flat Monge equations are successive integrable extensions of the Hilbert-Cartan equation. Many new examples are provided.

Lie Theorem via Rank 2 Distributions (Integration of PDE of Class ω = 1)

“…For instance, if ∆ is maximally symmetric non-Goursat distribution, then its deprolongation is flat in the sense of Tanaka [26], and so has the structure of successive integral extensions over the rank 2 distribution in R 5 with G 2 symmetry [3]. So we get the following theorem.…”

“…2 we discuss another more general method of integration of PDEs via integrable extensions (coverings), and relate this to the generalized symmetries. Notice that integrable extensions for rank 2 distributions were classified in [3], so their description in the symmetric cases reduces to purely algebraic questions.…”

“….). In this case by Cartan theorem [3,5] the system can be deprolonged, g i.e. there exists another manifoldM of dimensionμ = µ − 1 equipped with rank 2 distribution∆ such that ∆ = P(∆) is the prolongation.…”

“…These coverings of systems of ODEs (or distributions) were studied in [3] as they are useful in solving the system. Indeed a sequence of integrable extensions can decompose a given system into a sequence of 1st order scalar ODEs.…”

“…, I p that pull-back to first integrals of the system E. We can fix the values of I j and reduce the complexity of the system. g In [3] the growth vector of the weak derived flag was considered. However, when dim ∆ = 2, this makes no difference at the first three elements of the sequence (2, 3, x, .…”