On the Einstein-Weyl and conformal self-duality equations

TL;DR

This paper explores the integrability and explicit forms of Einstein-Weyl and conformal self-duality equations, connecting them to known integrable systems and providing new coordinate-based representations and Lax pairs.

Contribution

It presents explicit coordinate forms and Lax pairs for Einstein-Weyl and anti-self-dual conformal equations, linking Einstein-Weyl structures to the Manakov-Santini system.

Findings

Lorentzian Einstein-Weyl structures are locally solutions to the Manakov-Santini system.

Derived a system of two coupled third-order PDEs for anti-self-dual conformal structures in neutral signature.

Established explicit forms and Lax pairs for the equations in adapted coordinates.

Abstract

The equations governing anti-self-dual and Einstein-Weyl conformal geometries can be regarded as `master dispersionless systems' in four and three dimensions respectively. Their integrability by twistor methods has been established by Penrose and Hitchin. In this note we present, in specially adapted coordinate systems, explicit forms of the corresponding equations and their Lax pairs. In particular, we demonstrate that any Lorentzian Einstein-Weyl structure is locally given by a solution to the Manakov-Santini system, and we find a system of two coupled third-order scalar PDEs for a general anti-self-dual conformal structure in neutral signature.