  The most general cyclic representations of the quantum integrable tau_2-model
are analyzed. The complete characterization of the tau_2-spectrum (eigenvalues
and eigenstates) is achieved in the framework of Sklyanin's Separation of
Variables (SOV) method by extending and adapting the ideas first introduced in
[1, 2]: i) The determination of the tau_2-spectrum is reduced to the
classification of the solutions of a given functional equation in a class of
polynomials. ii) The determination of the tau_2-eigenstates is reduced to the
classification of the solutions of an associated Baxter equation. These last
solutions are proven to be polynomials for a quite general class of
tau_2-self-adjoint representations and the completeness of the associated Bethe
ansatz type equations is derived. Finally, the following results are derived
for the inhomogeneous chiral Potts model: i) Simplicity of the spectrum, for
general representations. ii) Complete characterization of the chiral Potts
spectrum (eigenvalues and eigenstates) and completeness of Bethe ansatz type
equations, for the self-adjoint representations of tau_2-model on the chiral
Potts algebraic curves.
