  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 Painleve I or II for
the dKP equation, but fails to pass the Painlev\'e test in higher dimensions.
The second generalisation of the dKP equation leads to a class of Einstein-Weyl
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 Einstein-Weyl spaces
belonging to this class, and depending on one arbitrary function of one
variable.
