A new Q-matrix in the Eight-Vertex Model

We construct a $Q$-matrix for the eight-vertex model at roots of unity for crossing parameter $\eta =2mK\mathrm{/}L$ with odd $L$, a case for which the existing constructions do not work. The new $Q$-matrix $\text{Q}$ depends as usual on the spectral parameter and also on a free parameter $t$. For $t=0$ $\text{Q}$ has the standard properties. For $t\mathrm{\ne }0$, however, it does not commute with the operator $S$ and not with itself for different values of the spectral parameter. We show that the six-vertex limit of $\text{Q}(v,t=i{K}^{\mathrm{\prime }}\mathrm{/}2)$ exists.