Second-order PDEs with integrable symbols

The aim of this project is to study nonlinear second order partial differential equations (PDEs) in three and four dimensions (3D, 4D) and establish integrability of certain classes of these PDEs based on some geometric properties of their principal symbols. In 3D, the geometric properties are the Einstein-Weyl geometry, which consists of an affine connection D, a conformal structure [g], and a covector ω. In 4D, the geometry is the self-dual geometry consisting of a vanishing (anti) self-dual part of the conformal structure [g]. Results attained provide partial classification in both dimensions, as well as conjectures for further investigation in future studies. In particular, for generic second-order PDEs in 3D, we show that ω, a covector appearing in the Einstein-Weyl structures, is expressible in terms of the equation. Also, we prove that the knowledge of ω and [g] provides a dispersionless Lax pair by an explicit formula. The rigidity conjecture proposes that for any generic second-order PDE with Einstein–Weyl property, all dependence on the 1-jet variables can be eliminated via a suitable contact transformation. As for the 4D case, we prove that the half-flatness requirement implies the Monge-Ampere property. Additionally we provide examples of nondegenerate integrable PDEs in 3D and 4D that are not contact equivalent to any translationally invariant equations.