  It is shown that the new Poisson brackets proposed in Part I of this work (J.
Math. Phys. 34, 5747(hep-th/9305133)) arise naturally in an extension of the
formal variational calculus incorporating divergences. The linear spaces of
local functionals, evolutionary vector fields, functional forms, multi-vectors
and differential operators become graded with respect to divergences. Bilinear
operations, such as action of vector fields onto functionals, commutator of
vector fields, interior product of forms and vectors and the Schouten-Nijenhuis
bracket are compatible with the grading. A definition of the adjoint graded
operator is proposed and skew-adjoint operators are constructed with the help
of boundary terms. Fulfilment of the Jacobi identity for the new Poisson
brackets is shown to be equivalent to vanishing of the Schouten-Nijenhuis
bracket for Poisson bivector with itself. The simple procedure for testing this
condition proposed by P. Olver is applicable with a minimal modification. It is
demonstrated that the second structure of the Korteweg--de Vries equation is
not Hamiltonian with respect to the new brackets.
