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

Abstract

The complex Monge-Amp\`ere equation $(CMA)$ in a two-component form is treated as a bi-Hamiltonian system. I present explicitly the first nonlocal symmetry flow in each of the two hierarchies of this system. An invariant solution of $CMA$ with respect to these nonlocal symmetries is constructed which, being a noninvariant solution in the usual sense, does not undergo symmetry reduction in the number of independent variables. I also construct the corresponding 4-dimensional anti-self-dual (ASD) Ricci-flat metric with either Euclidean or neutral signature. It admits no Killing vectors which is one of characteristic features of the famous gravitational instanton $K3$. For the metric with the Euclidean signature, relevant for gravitational instantons, I explicitly calculate the Levi-Civita connection 1-forms and the Riemann curvature tensor.