Nonlocal symmetry of CMA generates ASD Ricci-flat metric with no Killing vectors

Abstract

The complex Monge-Ampere equation $(CMA)$ in a two-component form is treated as a bi-Hamiltonian system. We present explicitly the first nonlocal symmetry flow in the hierarchy of this system. An invariant solution of $CMA$ with respect to this nonlocal symmetry is constructed which, being a noninvariant solution in the usual sense, does not undergo symmetry reduction in the number of independent variables. We also construct the corresponding 4-dimensional anti-self-dual (ASD) gravitational metric with either Euclidean or neutral signature. It admits no Killing vectors which is one of characteristic features of the famous gravitational instanton $K3$.