  We show how to lift solutions of Euclidean Einstein-Maxwell equations with
non-zero cosmological constant to solutions of eleven-dimensional supergravity
theory with non-zero fluxes. This yields a class of 11D metrics given in terms
of solutions to $SU(\infty)$ Toda equation. We give one example of a regular
solution and analyse its supersymmetry.
  We also analyse the integrability conditions of the Killing spinor equations
of N=2 minimal gauged supergravity in four Euclidean dimensions. We obtain
necessary conditions for the existence of additional Killing spinors,
corresponding to enhancement of supersymmetry. If the Weyl tensor is
anti-self-dual then the supersymmetric metrics satisfying these conditions are
given by separable solutions to the $SU(\infty)$ Toda equation. Otherwise they
are ambi-K\"ahler and are conformally equivalent to K\"ahler metrics of Calabi
type or to product metrics on two Riemann surfaces.
