AbstractEinstein–Weyl geometry is a triple $$({\mathbb {D}},g,\omega )$$
(
D
,
g
,
ω
)
where $${\mathbb {D}}$$
D
is a symmetric connection, [g] is a conformal structure and $$\omega $$
ω
is a covector such that $$\bullet $$
∙
connection $${\mathbb {D}}$$
D
preserves the conformal class [g], that is, $${\mathbb {D}}g=\omega g$$
D
g
=
ω
g
; $$\bullet $$
∙
trace-free part of the symmetrised Ricci tensor of $${\mathbb {D}}$$
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, covector $$\omega $$
ω
is a somewhat more mysterious object, recovered from the Einstein–Weyl conditions. We demonstrate that, for generic second-order PDEs (for instance, for all equations not of Monge–Ampère type), the covector $$\omega $$
ω
is also expressible in terms of the equation, thus providing an efficient ‘dispersionless integrability test’. The knowledge of g and $$\omega $$
ω
provides a dispersionless Lax pair by an explicit formula which is apparently new. Some partial classification results of PDEs with Einstein–Weyl characteristic conformal structure are obtained. A rigidity conjecture is proposed according to which 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.
Second-Order PDEs in 3D with Einstein–Weyl Conformal Structure
