In a previous paper, the first two authors classified complete Ricci-flat ALF
Riemannian 4-manifolds that are toric and Hermitian, but non-Kaehler. In this
article, we consider general Ricci-flat deformations of such spaces, assuming
only suitable fall-off conditions. Quite generally, we are able to show that
such a deformation must be Hermitian, and must carry a non-trivial Killing
vector field with fixed asymptotics. With mild additional hypotheses, we are
then able to show that the new Ricci-flat metric must in fact belong to the
family of previously classified metrics.Comment: 25 pages, LaTeX2e. Minor corrections. Many added reference