Summary: We review an algebraic method for constructing degenerate eigenvectors of the transfer matrix of the eight-vertex cyclic solid-on-solid lattice model (8V CSOS model), where the degeneracy increases exponentially with respect to the system size. We consider the elliptic quantum group \(E_{\tau,\eta}(\text{sl}2)\) at the discrete coupling constants: \(2N\;\eta = m_1 + im_2 \tau\), where \(N\), \(m_1\) and \(m_2\) are integers. Then we show that degenerate eigenvectors of the transfer matrix of the six-vertex model at roots of unity in the sector \(S^Z \equiv 0\;(\mod N\)) are derived from those of the 8V CSOS model, through the trigonometric limit. They are associated with the complete N strings. From the result we see that the dimension of a given degenerate eigenspace in the sector \(S^Z \equiv 0\;(\mod N\)) of the six-vertex model at \(N\)th roots of unity is given by \(2^{2S^Z_{max}/N}\), where \(S^Z_{max}\) is the maximal value of the total spin operator \(S^Z\) in the degenerate eigenspace.