Summary: We discuss an algebraic method for constructing eigenvectors of the transfer matrix of the eight-vertex model at the discrete coupling parameters. We consider the algebraic Bethe ansatz of the elliptic quantum group \(E_{\tau,\eta} (sl_2)\) for the case where the parameter \(\eta\) satisfies \(2N\eta= m_1+m_2\tau\) for arbitrary integers \(N\), \(m_1\) and \(m_2\). When \(m_1\) or \(m_2\) is odd, the eigenvectors thus obtained have not been discussed previously. Furthermore, we construct a family of degenerate eigenvectors of the \(XYZ\) spin chain, some of which are shown to be related to the \(sl_2\) loop algebra symmetry of the \(XXZ\) spin chain. We show that the dimension of some degenerate eigenspace of the \(XYZ\) spin chain on \(L\) sites is given by \(N2^{L/N}\), if \(L/N\) is an even integer. The construction of eigenvectors of the transfer matrices of some related interaction around a face models is also discussed.