The quadric ansatz for the mn-dispersionless KP equation, and supersymmetric Einstein–Weyl spaces

Abstract

We consider two multi-dimensional generalisations of the dispersionless Kadomtsev–Petviashvili (dKP) equation, both allowing for arbitrary dimensionality, and non-linearity. For one of these generalisations, we characterise all solutions which are constant on a central quadric. The quadric ansatz leads to a second order ODE which is equivalent to Painlevé I or II for the dKP equation, but fails to pass the Painlevé test in higher dimensions. The second generalisation of the dKP equation leads to a class of Einstein–Weyl (EW) structures in an arbitrary dimension, which is characterised by the existence of a weighted parallel vector field, together with further holonomy reduction. We construct and characterise an explicit new family of EW spaces belonging to this class, and depending on one arbitrary function of one variable.