A one-root Bethe equation need not produce a complete set of vectors

Let $\mathcal{H}=\mathbb{C}{e}_0\oplus \mathbb{C}{e}_1$ be the direct sum of two counit representations and set $\mathrm{\Omega }=e_0$. Applying the counit on each summand gives $t_{ij}(u)={\delta }_{ij}{I}_{\mathcal{H}}$, hence $A=D=I$ and $B=C=0$. These constant operator families are regular on the common spectral domain $\mathcal{D}=\mathbb{C}$ and satisfy the pointwise RTT identities. Thus the pseudovacuum conditions hold there with $a(u)=d(u)=1$, and the one-root equation in [F2] holds for every $v\in \mathcal{D}$.

For every $k\ge 1$ and parameters $v_1,\dots ,v_k$, the rightmost $B(v_k)$ sends $\mathrm{\Omega }$ to zero, so $B(v_1)\cdots B(v_k)\mathrm{\Omega }=0$. The empty product is $e_0$. Hence all Bethe vectors from this pseudovacuum span $\mathbb{C}{e}_0$, a proper subspace because $e_1$ is linearly independent from $e_0$. Thus the one-root equation by itself does not guarantee completeness.