In this talk, it is shown that, under a symmetry assumption
the equations governing a generic anti-self-dual conformal structure in
four dimensions can be explicitly reduced to the Manakov-Santini system,
which determines a generic three-dimensional Lorentzian Einstein-Weyl
structure, using a simple transformation. Then, motivated by the dKP
Einstein-Weyl structure, two generalisations of the dKP (dispersionless
Kadomtsev-Petviashvili) equation to higher dimensions are discussed. For
one generalisation, its (non)integrability is investigated by
constructing solutions constant on central quadrics. Another
generalisation determines a class of Einstein-Weyl structures in n+2
dimensions, for which an explicit local expression for a subclass is
obtained.