General heavenly equation governs anti-self-dual gravity

Abstract

We show that the general heavenly equation, suggested recently by Doubrov and Ferapontov (2010 arXiv:0910.3407v2 [math.DG]), governs anti-self-dual (ASD) gravity. We derive ASD Ricci-flat vacuum metric governed by the general heavenly equation, null tetrad and basis of 1-forms for this metric. We present algebraic exact solutions of the general heavenly equation as a set of zeros of homogeneous polynomials in independent and dependent variables. A real solution is obtained for the case of a neutral signature.