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We study the eigenvector problem in homogeneous superintegrable $N$-statechiral Potts model (CPM) by the symmetry principal. Using duality symmetry and(spin-)inversion in CPM, together with Onsager-algebra symmetry and$sl_2$-loop-algebra symmetry of the superintegrable $\tau^{(2)}$-model, weconstruct the complete $k'$-dependent CPM-eigenvectors in the local spin basisfor an arbitrary Onsager sector. In this paper, we present the completeclassification of quantum numbers of superintegrable $\tau^{(2)}$-model.Accordingly, there are four types of sectors. The relationships among Onsagersectors under duality and inversion, together with their Bethe roots andCPM-eigenvectors, are explicitly found. Using algebraic-Bethe-ansatz techniquesand duality of CPM, we construct the Bethe states and the Fabricius-McCoycurrents of the superintegrable $\tau^{(2)}$-model through its equivalentspin-$\frac{N-1}{2}$-XXZ chain. The $\tau^{(2)}$-eigenvectors in a sector arederived from the Bethe state and the $sl_2$-product structure determined by theFabricius-McCoy current of the sector. From those $\tau^{(2)}$-eigenvectors,the $k'$-dependence of CPM state vectors in local-spin-basis form is obtainedby the Onsager-algebra symmetry of the superintegrable chiral Potts quantumchain.

Eigenvectors of an Arbitrary Onsager Sector in Superintegrable $\tau^{(2)}$-model and Chiral Potts Model