  Classical $W$-algebras in higher dimensions are constructed. This is achieved
by generalizing the classical Gel'fand-Dickey brackets to the commutative limit
of the ring of classical pseudodifferential operators in arbitrary dimension.
These $W$-algebras are the Poisson structures associated with a higher
dimensional version of the Khokhlov-Zabolotskaya hierarchy (dispersionless
KP-hierarchy). The two dimensional case is worked out explicitly and it is
shown that the role of Diff$S(1)$ is taken by the algebra of generators of
local diffeomorphisms in two dimensions.
