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.