I will present two results in connection with anti-self-
dual equations in four dimensions. Firstly, an affine sphere equation
is shown to be a symmetry reduction of the anti-self-dual Yang-Mills
equation, which confirms its integrability by twistor
method. Secondly, a generalization of the dKP equation which
determines a family of Einstein-Weyl structures in an arbitrary
dimension will be discussed. The dKP equation itself is integrable, and
can be realised as a reduction of the anti-self-dual conformal
equation. Although, the generalised equation is not integrable in a
dimension greater than three, an extended version of the quadric ansatz
method will be presented as an attempt to find solutions of the equation.