Einstein–Weyl geometry is a triple (D, g, ω) where D is a symmetric connection,[g] is a
conformal structure and ω is a covector such that∙ connection D preserves the conformal
class [g], that is, D g= ω g;∙ trace-free part of the symmetrised Ricci tensor of D vanishes.
Three-dimensional Einstein–Weyl structures naturally arise on solutions of second-order
dispersionless integrable PDEs in 3D. In this context,[g] coincides with the characteristic
conformal structure and is therefore uniquely determined by the equation. On the contrary …

• Einstein-Weyl property is equivalent to the existence of a dispersionless Lax pair in λ-
dependent commuting vector fields (Calderbank, Kruglikov, 2016).• Rigidity conjecture: a
'generic'(for example, non-Monge-Ampere) second-order PDE satisfying Einstein-Weyl
property can be reduced to a dispersionless Hirota form via a suitable contact
transformation.