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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 theformal variational calculus incorporating divergences. The linear spaces oflocal functionals, evolutionary vector fields, functional forms, multi-vectorsand differential operators become graded with respect to divergences. Bilinearoperations, such as action of vector fields onto functionals, commutator ofvector fields, interior product of forms and vectors and the Schouten-Nijenhuisbracket are compatible with the grading. A definition of the adjoint gradedoperator is proposed and skew-adjoint operators are constructed with the helpof boundary terms. Fulfilment of the Jacobi identity for the new Poissonbrackets is shown to be equivalent to vanishing of the Schouten-Nijenhuisbracket for Poisson bivector with itself. The simple procedure for testing thiscondition proposed by P. Olver is applicable with a minimal modification. It isdemonstrated that the second structure of the Korteweg--de Vries equation isnot Hamiltonian with respect to the new brackets.

Boundary values as Hamiltonian variables. II. Graded structures