Anti-self-dual Riemannian metrics without Killing vectors: can they be realized on K3?

Explicit Riemannian metrics with Euclidean signature and anti-self-dual curvature that do not admit any Killing vectors are presented. The metric and the Riemann curvature scalars are homogeneous functions of degree zero in a single real potential and its derivatives. The solution for the potential is a sum of exponential functions which suggests that for the choice of a suitable domain of coordinates and parameters it can be the metric on a compact manifold. Then, by the theorem of Hitchin, it could be a class of metrics on K3, or on surfaces whose universal covering is K3.