Note that this kind of staggering mechanism for the stationary state has already appeared in [@ProsenB].

For an spectral parameter, the stationary state is given by $${|{\cal S}\rangle}= \frac{1}{Z_L} {\langle\!\langle}W|\mathbf{A}(\kappa)\otimes\mathbf{A}(-\kappa)\otimes\dots\otimes\mathbf{A}(\kappa)|V{\rangle\!\rangle}$$ where $Z_L= Z_L(\kappa,-\kappa,\dots,\kappa)$.

We recall also that the construction of the stationary state exposed in this subsection holds only for models with open boundary conditions. For models with periodic boundary conditions, we would need to adapt the construction of the stationary state $${|{\cal S}\rangle}_N = \frac{1}{Z_{L,N}} \pi_N\Big[
 tr \left(\mathbf{A}(\kappa)\otimes\mathbf{A}\left(\frac{1}{\kappa}\right)\otimes\dots\otimes \mathbf{A}\left(\frac{1}{\kappa}\right)\right)\Big],$$ where $tr(\cdot)$ denotes a trace operator (*i.e* satisfying the cyclic property) on the auxiliary space of the matrix ansatz algebra. $\pi_N$ is the projector (in the physical space of configurations) on the sector with $N$ particles on the lattice. Indeed with periodic boundary conditions, the number of particles in the system is often conserved and the configuration space can be decomposed into different sectors (labeled by the number of particles) that are stable under the action of the Markov matrix. $Z_{L,N}$ is the appropriate normalization (so that the sum of the components of ${|{\cal S}\rangle}_N$ is $1$).

To ease the presentation, and to avoid dealing with these different sectors, we will focus on systems with open boundary conditions in the 