  In [1] an integrable quantum model was introduced and a class of its cyclic
representations was proven to define lattice regularizations of the Sine-Gordon
model. Here, we analyze general cyclic representations of this integrable
quantum model by extending the spectrum construction introduced in [2] in the
framework of the Separation of Variables (SOV) of Sklyanin. We show that as in
[1] also for general representations, the transfer matrix spectrum (eigenvalues
and eigenstates) is completely characterized in terms of polynomial solutions
of an associated functional Baxter equation. Moreover, we prove that the method
here developed has two fundamental built-in features: i) the completeness of
the set of the transfer matrix eigenstates constructed from the solutions of
the associated Bethe ansatz equations, ii) the existence and complete
characterization of the Baxter Q-operator.
