  Recently Strachan introduced a Moyal algebraic deformation of selfdual
gravity, replacing a Poisson bracket of the Plebanski equation by a Moyal
bracket. The dressing operator method in soliton theory can be extended to this
Moyal algebraic deformation of selfdual gravity. Dressing operators are defined
as Laurent series with coefficients in the Moyal (or star product) algebra, and
turn out to satisfy a factorization relation similar to the case of the KP and
Toda hierarchies. It is a loop algebra of the Moyal algebra (i.e., of a
$W_\infty$ algebra) and an associated loop group that characterize this
factorization relation. The nonlinear problem is linearized on this loop group
and turns out to be integrable.
