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We discuss an algebraic method for constructing eigenvectors of the transfermatrix of the eight vertex model at the discrete coupling parameters. Weconsider the algebraic Bethe ansatz of the elliptic quantum group $E_{\tau,\eta}(sl_2)$ for the case where the parameter $\eta$ satisfies $2 N \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 spinchain, some of which are shown to be related to the $sl_2$ loop algebrasymmetry of the XXZ spin chain. We show that the dimension of some degenerateeigenspace of the XYZ spin chain on $L$ sites is given by $N 2^{L/N}$, if $L/N$is an even integer. The construction of eigenvectors of the transfer matricesof some related IRF models is also discussed.Comment: 19 pages, no figure (revisd version with three appendices

journal of physics. a, mathematical and general/journal of physics. a, mathematical and general

Construction of some missing eigenvectors of the XYZ spin chain at the discrete coupling constants and the exponentially large spectral degeneracy of the transfer matrix