  In the early 1970's, Gelfand, Kalinin and Fuks found an exotic characteristic
class of degree 7 in the Gelfand-Fuks cohomology of the Lie algebra of formal
Hamiltonian vector fields on the plane. We prove that this cohomology class can
be decomposed as a product of a certain leaf cohomology class of degree 5 and
the transverse symplectic class. This is similar to the well known
factorization of the Godbillon-Vey class for codimension n foliations. We also
interpret the characteristic classes of transversely symplectic foliations
introduced by Kontsevich in terms of the known classes and prove non-triviality
for some of them.
