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The most general cyclic representations of the quantum integrable tau_2-modelare analyzed. The complete characterization of the tau_2-spectrum (eigenvaluesand eigenstates) is achieved in the framework of Sklyanin's Separation ofVariables (SOV) method by extending and adapting the ideas first introduced in[1, 2]: i) The determination of the tau_2-spectrum is reduced to theclassification of the solutions of a given functional equation in a class ofpolynomials. ii) The determination of the tau_2-eigenstates is reduced to theclassification of the solutions of an associated Baxter equation. These lastsolutions are proven to be polynomials for a quite general class oftau_2-self-adjoint representations and the completeness of the associated Betheansatz type equations is derived. Finally, the following results are derivedfor the inhomogeneous chiral Potts model: i) Simplicity of the spectrum, forgeneral representations. ii) Complete characterization of the chiral Pottsspectrum (eigenvalues and eigenstates) and completeness of Bethe ansatz typeequations, for the self-adjoint representations of tau_2-model on the chiralPotts algebraic curves.

The tau_2-model and the chiral Potts model revisited: completeness of Bethe equations from Sklyanin's SOV method