  In this paper, we introduce the notion of cyclic Frobenius algebras
(CF-algebras). Canonical structures of CF-algebras exist on associative and
Poisson algebras. It turns out that the modern theory of integrable systems
yields non-trivial examples of CF-algebras. In the theory of the KdV hierarchy
there is a structure of CF-algebra which leads to explicit expressions for the
first integrals of the N-th Novikov hierarchy, suitable for classical,
non-Abelian and quantum cases.
