General heavenly equation governs anti-self-dual gravity

Abstract

We show that the general heavenly equation, suggested recently by Doubrov and Ferapontov \cite{fer}, 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 neutral signature.

LaTeX2e, 17 pages. Important additions and corrections made. A new section added on derivation of Lax pair

References in corpus (3)

• On the integrability of symplectic Monge-Ampére equations
• On a class of second-order PDEs admitting partner symmetries
• Recursions of Symmetry Orbits and Reduction without Reduction

Cited by in corpus (5)

• A Simple Construction of Recursion Operators for Multidimensional Dispersionless Integrable Systems
• Recursion operators and bi-Hamiltonian structure of the general heavenly equation
• Recursions of Symmetry Orbits and Reduction without Reduction
• Anti-self-dual gravity from asymmetric heavenly equation standpoint
• On some exact solutions of heavenly equations in four dimensions