  For any generalized Frobenius manifold with non-flat unity, we construct a
bihamiltonian integrable hierarchy of hydrodynamic type which is an analogue of
the Principal Hierarchy of a Frobenius manifold. We show that such an
integrable hierarchy, which we also call the Principal Hierarchy, possesses
Virasoro symmetries and a tau structure, and the Virasoro symmetries can be
lifted to symmetries of the tau-cover of the integrable hierarchy. We derive
the loop equation from the condition of linearization of actions of the
Virasoro symmetries on the tau function, and construct the topological
deformation of the Principal Hierarchy of a generalized semisimple Frobenius
manifold with non-flat unity. We also give two examples of generalized
Frobenius manifolds with non-flat unity which are shown to be closely related
to the well-known integrable hierarchies: the Volterra hierarchy, the
q-deformed KdV hierarchy and the Ablowitz-Ladik hierarchy.
