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.

A dressing scheme applicable to Dunajski equation is developed. Simple example of constructing solutions in terms of implicit functions is considered. Dunajski equation hierarchy is described, its Lax-Sato form is presented. Dunajsky equation hierarchy is characterised by conservation of three-dimensional volume form, in which a spectral variable is taken into account. Some reductions of Dunajsky equation hierarchy, including waterbag-type reduction, are studied. 1 Dunajski equation Dunajski equation [1] is a representative of the class integrable sytems arising in the context of complex relativity [2]-[6]. It is closely connected to the selebrated Plebański heavenly equations [2] and in some sense generalizes them. It describes anti-self-dual null-Kähler structures. In [1] it was shown that all null-Kähler metrics (signature (2,2)) locally admit a canonical Plebański form g = dwdx + dzdy − Θxxdz 2 − Θyydw 2 + 2Θxydwdz. (1) The conformal anti-self-duality (ASD) condition leads to Dunajski equation

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Integrable (2+1) dimensional Manakov-Santini system connected with commutation of 2-dimensional vector fields is considered. Lax-Sato form and generating equation of the corresponding hierarchy are introduced. Waterbag reduction for Manakov-Santini hierarchy is defined and equations of the reduced hierarchy are derived. 1

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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 and GL(2, R) structures.

We use deformations of Lie algebra homomorphisms to construct deformations of dispersionless integrable systems arising as symmetry reductions of anti–self–dual Yang–Mills equations with a gauge group Diff(S 1).