Abstract
The complex Monge–Ampère (CMA) equation 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 the CMA equation 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 four-dimensional anti-self-dual Ricci-flat metric with either the Euclidean or neutral signature. It admits no Killing vectors, which is one of the 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 one-forms and the Riemann curvature tensor.
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