  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.
