An interpolating dispersionless integrable system

We introduce a dispersionless integrable system which interpolates between the dispersionless Kadomtsev-Petviashvili equation and the hyper-CR equation. The interpolating system arises as a symmetry reduction of the anti-self-dual Einstein equations in (2, 2) signature by a conformal Killing vector whose self-dual derivative is null. It also arises as a special case of the Manakov-Santini integrable system. We discuss the corresponding Einstein-Weyl structures.