equation in a matrix form, introducing a Markov matrix and a probability vector. The first step toward this goal is to associate to each value $\tau=0,\dots,s$ of the local state variable with a local basis vector of $\cV={{\mathbb C}}^{s+1}$ $${{|\tau\rangle}}=(\underbrace{0,\dots,0}_{\tau},1,\underbrace{0,\dots,0}_{s-\tau})^t.$$ It allows us to define a probability vector $${{|\cP_t\rangle}} = \sum_{\bm{\tau}} \cP_t(\bm{\tau}) {{|\bm{\tau}\rangle}},$$ where we have constructed a basis vector of $\cV^{\otimes L}$ associated to each configuration $\bm{\tau}$ as tensor product of elementary vectors $${{|\bm{\tau}\rangle}} = {{|\tau_1\rangle}} \otimes {{|\tau_2\rangle}} \otimes \cdots \otimes {{|\tau_L\rangle}}.$$ The time evolution of the probability vector can be encoded in a master equation that takes slightly different forms depending if the stochastic model describes *continuous-time* or *discrete-time* dynamics. For *continuous-time* dynamics, the master equation reads $$\frac{d{{|\cP_{t}\rangle}}}{dt} = M{{|\cP_t\rangle}},$$ where $M$ is a *continuous-time* Markov matrix (acting on the space $\cV^{\otimes L}$) with non-negative off-diagonal entries and whose sum of the entries of each column vanishes.

In the case of a *discrete-time* dynamics, the master equation reads $${{|\cP_{t+1}\rangle}} = M{{|\cP_t\rangle}},$$ where $M$ is a *discrete-time* Markov matrix with non-negative entries whose sum over each column is one. The present paper will be essentially devoted to the construction of integrable *discrete-time* Markov matrices $M$ from the transfer matrix 