Second-Order PDEs in 3D with Einstein–Weyl Conformal Structure

Einstein-Weyl geometry is a triple (D,g,w), where D symmetric connection, [g] conformal structure and w covector such that: (i) connection preserves the class [g], that is, Dg=wg; (ii) trace-free part of symmetrised Ricci tensor vanishes. Three-dimensional structures arise naturally on solutions second-order dispersionless integrable PDEs in 3D. In this context, coincides with characteristic therefore uniquely determined by equation. On contrary, somewhat more mysterious object, recovered from conditions. We demonstrate that, for generic (for instance, all equations not Monge-Ampere type), also expressible terms equation, thus providing an efficient integrability test. The knowledge g provides Lax pair explicit formula which apparently new. Some partial classification results are obtained. A rigidity conjecture proposed according to any PDE property, dependence 1-jet variables can be eliminated via suitable contact transformation.