  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 $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 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 $N 2^{L/N}$, if $L/N$
is an even integer. The construction of eigenvectors of the transfer matrices
of some related IRF models is also discussed.
