  Making use of the extended flux homomorphism on the group of
symplectomorphisms of a closed oriented surface of genus at least 2, we
introduce new characteristic classes of foliated surface bundles with
symplectic, equivalently area-preserving, total holonomy. These characteristic
classes are stable with respect to the genus and we show that they are highly
non-trivial. We also prove that the second homology of the group of Hamiltonian
symplectomorphisms equipped with the discrete topology is very large.
