  We prove that the existence of a dispersionless Lax pair with spectral
parameter for a nondegenerate hyperbolic second order partial differential
equation (PDE) is equivalent to the canonical conformal structure defined by
the symbol being Einstein-Weyl on any solution in 3D, and self-dual on any
solution in 4D. The first main ingredient in the proof is a characteristic
property for dispersionless Lax pairs. The second is the projective behaviour
of the Lax pair with respect to the spectral parameter. Both are established
for nondegenerate determined systems of PDEs of any order. Thus our main result
applies more generally to any such PDE system whose characteristic variety is a
quadric hypersurface.

Analysis of PDEs;Differential Geometry;Exactly Solvable and Integrable Systems