  In this paper, we extend purely non-local Hamiltonian formalism to a class of
Riemannian F-manifolds, without assumptions on the semisimplicity of the
product $\circ$ or on the flatness of the connection $\nabla$. In the flat case
we show that the recurrence relations for the principal hierarchy can be
re-interpreted using a local and purely non-local Hamiltonian operators and in
this case they split into two Lenard-Magri chains, one involving the even
terms, the other involving the odd terms. Furthermore, we give an elementary
proof that the Kohno property and the $\vee$-system condition are equivalent
under suitable conditions and we show how to associate a purely non-local
Hamiltonian structure to any $\vee$-system, including degenerate ones.
