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.