CONTINUUM LIE-ALGEBRAS AND THE SCALAR-FLAT KAHLER SURFACES

We show that the equations derived by LeBrun to describe scalar-flat K ahler surfaces with a conformal S1-action are the continuous long wave approximation for the Toda-type system associated with the semi-direc t sum of the algebra A(r) and the commutive multiplet which transforms according to the adjoint representation of A(r). Moreover, we display the corresponding continuum Lie algebra, on which the Maurer-Cartan 1 -form (written explicitly) takes values for the equations under consid eration, the improved energy-momentum tensor, etc.