  We demonstrate how a combination of our recently developed methods of partner
symmetries, symmetry reduction in group parameters and a new version of the
group foliation method can produce noninvariant solutions of complex
Monge-Amp\`ere equation (CMA) and provide a lift from invariant solutions of
CMA satisfying Boyer-Finley equation to non-invariant ones. Applying these
methods, we obtain a new noninvariant solution of CMA and the corresponding
Ricci-flat anti-self-dual Einstein-K\"ahler metric with Euclidean signature
without Killing vectors, together with Riemannian curvature two-forms. There
are no singularities of the metric and curvature in a bounded domain if we
avoid very special choices of arbitrary functions of a single variable in our
solution. This metric does not describe gravitational instantons because the
curvature is not concentrated in a bounded domain.
