  Four-dimensional quaternion-Kahler metrics, or equivalently self-dual
Einstein spaces M, are known to be encoded locally into one real function h
subject to Przanowski's Heavenly equation. We elucidate the relation between
this description and the usual twistor description for quaternion-Kahler
spaces. In particular, we show that the same space M can be described by
infinitely many different solutions h, associated to different complex (local)
submanifolds on the twistor space, and therefore to different (local)
integrable complex structures on M. We also study quaternion-Kahler
deformations of M and, in the special case where M has a Killing vector field,
show that the corresponding variations of h are related to eigenmodes of the
conformal Laplacian on M. We exemplify our findings on the four-sphere S^4, the
hyperbolic plane H^4 and on the "universal hypermultiplet", i.e. the
hypermultiplet moduli space in type IIA string compactified on a rigid
Calabi-Yau threefold.
