2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/187210The 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 $sdiff(Σ^2)$ used in the study of the undeformed, or dispersionless, equations.11 pages, LaTeX. To appear: J. Phys. AHigh Energy Physics - TheoryExactly Solvable and Integrable Systemstext