Second-order PDEs in 3D with Einstein-Weyl conformal structure

I will discuss a general class of second-order PDEs in 3D whose characteristic conformal structure satisfies the Einstein-Weyl conditions on every solution.
This property is known to be equivalent to the existence of a dispersionless Lax pair, as well as to other equivalent definitions of dispersionless integrability.
I will demonstrate that (a) the Einstein-Weyl conditions can be viewed as an efficient contact-invariant test of dispersionless integrability, (b) show some partial classification results, and (c) formulate a rigidity conjecture according to which any second-order PDE with Einstein-Weyl conformal structure can be reduced to a dispersionless Hirota form via a suitable contact transformation.
Based on joint work with S. Berjawi, B. Kruglikov, V. Novikov.