  In our recent paper [DSK11] we computed the dimension of the variational
Poisson cohomology for any quasiconstant coefficient matrix differential
operator K of arbitrary order with invertible leading coefficient, provided
that the algebra of differential functions is normal and is an algebra over a
linearly closed differential field. In the present paper we show that, for K
skewadjoint, this cohomology, viewed as a Z-graded Lie superalgebra, is
isomorphic to the finite dimensional Lie superalgebra of Hamiltonian vector
fields over a Grassman algebra. We also prove that the subalgebra of
`essential' variational Poisson cohomology, consisting of classes vanishing on
the Casimirs of K, is zero. This vanishing result has applications to the
theory of bi-Hamiltonian structures and their deformations. At the end of the
paper we consider also the translation invariant case.
