Generalized symmetries for the sDiff(2) Toda equation

Self-dual Einstein metrics which admit one (rotational) symmetry vector are determined by solutions of the sDiff(2) Toda equation, which has also been studied in a variety of other physical contexts. Non-trivial solutions are difficult to obtain, with considerable effort in that direction recently. Therefore much effort has been involved with determining solutions with symmetries, and also with a specific lack of symmetries. The contact symmetries have been known for some time, and form an infinite-dimensional Lie algebra over the jet bundle of the equation. Generalizations of those symmetries to include derivatives of arbitrary order are often referred to as higher- or generalized-symmetries. Those symmetries are described, with the unexpected result that their existence also requires prolongations to "potentials" for the original dependent variables for the equation: potentials which are generalizations of those already usually introduced for this equation. Those prolongations are described, and the prolongations of the commutators for the symmetry generators are created. The generators so created form an infinite-dimensional, Abelian, Lie algebra, defined over these prolongations.