Anti-self-dual gravitational metrics determined by the modified heavenly equation

In paper Doubrov and Ferapontov (2010) on the classification of integrable complex Monge-Ampère equations, the modified heavenly (MH) equation of Dubrov and Ferapontov is one of canonical equations. It is well known that solutions of the first and second heavenly equations of Plebañski (1975) and those of the Husain equation in Husain (1994) provide potentials for anti-self-dual (ASD) Ricci-flat vacuum metrics. For another canonical equation, the general heavenly equation of Dubrov and Ferapontov (2010), we had constructed in Malykh and Sheftel (2011) ASD Ricci-flat metric governed by this equation. Thus, the modified heavenly equation remains the only one in the list of canonical equations in Doubrov and Ferapontov (2010) for which such a metric is missing so far. Our aim here is to construct null tetrad of vector fields, coframe 1-forms and ASD Ricci-flat metric for the latter equation. We study reality conditions and signature for the resulting metric. As an example, we obtain a multi-parameter cubic solution of the MH equation which yields a family of metrics with the above properties. Riemann curvature 2-forms are also explicitly presented for the cubic solution.