  We consider the pairs of general weakly non-local Poisson brackets of
Hydrodynamic Type (Ferapontov brackets) and the corresponding integrable
hierarchies. We show that under the requirement of non-degeneracy of the
corresponding "first" pseudo-Riemannian metric $g_{(0)}^{\nu\mu}$ and also some
non-degeneracy requirement for the nonlocal part it is possible to introduce a
"canonical" set of "integrable hierarchies" based on the Casimirs, Momentum
functional and some "Canonical Hamiltonian functions". We prove also that all
the "Higher" "positive" Hamiltonian operators and the "negative" symplectic
forms have the weakly non-local form in this case. The same result is also true
for "negative" Hamiltonian operators and "positive" Symplectic structures in
the case when both pseudo-Riemannian metrics $g_{(0)}^{\nu\mu}$ and
$g_{(1)}^{\nu\mu}$ are non-degenerate.
