approach. We will also be interested in computing exactly in a matrix product form the stationary state[^2] [^3] ${|{\cal S}\rangle}$, satisfying $$\label{eq:steady}
 M {|{\cal S}\rangle}= {|{\cal S}\rangle}.$$

#### Periodic boundary conditions.

The building block of integrable models with periodic boundary conditions is a matrix $R(z)$ depending on a spectral parameter $z$ and acting on two components $\cV\otimes\cV$ of the tensor space $\cV$ (*i.e* on two sites of the lattice).

The key equation for integrability in the periodic boundary case is the Yang-Baxter equation $$\label{eq:YBE}
 R_{12}\left(\frac{z_1}{z_2}\right) R_{13}\left(\frac{z_1}{z_3}\right) R_{23}\left(\frac{z_2}{z_3}\right) = 
 R_{23}\left(\frac{z_2}{z_3}\right) R_{13}\left(\frac{z_1}{z_3}\right) R_{12}\left(\frac{z_1}{z_2}\right).$$ The previous equation holds in $\cV\otimes\cV\otimes\cV$. The subscripts indicate on which components of the tensor space the matrix $R$ is acting non-trivially. For instance $R_{12}(z)=R(z)\otimes 1$, $R_{23}(z)=1 \otimes R(z)$,... We also require that the matrix $R(z)$ satisfies further properties. We are in this paper interested in constructing integrable discrete time Markov matrices, *i.e* matrices with non-negative entries whose sum on each column is equal to one. We will therefore consider $R$-matrices satisfying the Markovian property $$\label{eq:markovian}
 {{\langle\sigma|}} \otimes {{\langle\sigma|}} R(z) = {{\langle\sigma|}} \otimes {{\langle\sigma|}}$$ where ${{\langle\sigma|}} = \sum_{v=0}^{s}{{\langlev|}}$ is the row vector with all entries equal to one. The latter property is nothing else than imposing that the sum 