  The Boyer-Finley equation, or $SU(\infty)$-Toda equation is both a reduction
of the self-dual Einstein equations and the dispersionlesslimit of the
$2d$-Toda lattice equation. This suggests that there should be a dispersive
version of the self-dual Einstein equation which both contains the Toda lattice
equation and whose dispersionless limit is the familiar self-dual Einstein
equation. Such a system is studied in this paper. The results are achieved by
using a deformation, based on an associative $\star$-product, of the algebra
$sdiff(\Sigma^2)$ used in the study of the undeformed, or dispersionless,
equations.
