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Eigenfunctions are shown to constitute privileged coordinates of self-dualEinstein spaces with the underlying governing equation being revealed as thegeneral heavenly equation. The formalism developed here may be used to linkalgorithmically a variety of known heavenly equations. In particular, theclassical connection between Plebanski's first and second heavenly equations isretrieved and interpreted in terms of eigenfunctions. In addition, connectionswith travelling wave reductions of the recently introduced TED equation whichconstitutes a 4+4-dimensional integrable generalisation of the general heavenlyequation are found. These are obtained by means of (partial) Legendretransformations. As a particular application, we prove that a large class ofself-dual Einstein spaces governed by a compatible system of dispersionlessHirota equations is genuinely four-dimensional in that the (generic) metrics donot admit any (proper or non-proper) conformal Killing vectors. Thisgeneralises the known link between a particular class of self-dual Einsteinspaces and the dispersionless Hirota equation encoding three-dimensionalEinstein-Weyl geometries.

Self-dual Einstein spaces and the general heavenly equation. Eigenfunctions as coordinates