THE DISPERSIVE SELF-DUAL EINSTEIN EQUATIONS AND THE TODA LATTICE

The Boyer-Finley equation, or SU(infinity) Toda equation is both a red uction of the self-dual Einstein equations and the dispersionless limi t of the 2D Toda lattice equation. This suggests that there should be a dispersive version of the self-dual Einstein equation which both con tains 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 a ssociative star-product, of the algebra sdiff(Sigma 2) used in the stu dy of the undeformed, or dispersionless, equations.