  We discuss hamiltonian structures of the Gelfand-Dorfman complex of
projectable vector fields and differential forms on a foliated manifold. Such a
structure defines a Poisson structure on the algebra of foliated functions, and
embeds the given foliation into a larger, generalized foliation with
presymplectic leaves. In a so-called tame case, the structure is induced by a
Poisson structure of the manifold. Cohomology spaces and classes relevant to
geometric quantization are also considered.
