[@sklyanin]. The discrete time dynamics that we obtain for both periodic and open boundary conditions appears to be two-step Floquet dynamics. Note that another approach has been developed recently in [@FloquetInt] to construct integrable continuous time Floquet dynamics for quantum systems from the transfer matrix formalism.

#### Periodic boundary conditions.

To define our process on the periodic lattice, we need to consider a lattice with an even number of sites $L$. We fix $z_1=z_3=\dots=z_{L-1}=\frac{1}{\kappa}$ and $z_2=z_4=\dots=z_L=\kappa$. The inhomogeneous transfer matrix then takes the following staggered form $$\label{eq:transfer_matrix_periodic}
 t(z)=tr_0\left( R_{0L}\left(\frac{z}{\kappa}\right)R_{0L-1}\left(z\kappa\right) \dots R_{02}\left(\frac{z}{\kappa}\right)R_{01}\left(z\kappa\right) \right).$$ Note that a similar staggered construction already appeared in a monodromy matrix in [@FaddeevV]. Let us introduce the local Markovian operator (acting on two sites) $$\label{eq:definition_U}
 U = \check R(\kappa^2)$$ where $\check R(z)=P.R(z)$. Note that $U$ also fulfill the Markovian property ${{\langle\sigma|}}U={{\langle\sigma|}}$. A straightforward computation[^6] yields $$t\left(\frac{1}{\kappa}\right)= P_{1L}P_{1L-1}\dots P_{13}P_{12} \times U_{L-1,L}^{-1} U_{L-3,L-2}^{-1} \dots U_{1,2}^{-1}$$ and $$t(\kappa) = P_{1L}P_{1L-1}\dots P_{13}P_{12} \times U_{L-2,L-1}U_{L-4,L-3} \dots U_{2,3}U_{L,1}.$$ We can thus define a Markov matrix $M$ in the following way $$\label{eq:Markov_matrix_periodic}
 M=t\left(\frac{1}{\kappa}\right)^{-1}t(\kappa) = {{\mathbb U}}^{o} {{\mathbb U}}^{e}$$ with[^7] $$\label{eq:Floquet_operators_periodic}
 {{\mathbb U}}^{o} = \prod_{k=1}^{L/2} U_{2k-1,2k} \quad \mbox{and} \quad {{\mathbb U}}^{e} = \prod_{k=1}^{L/2} U_{2k,2k+1}.$$ The property that expression together with fulfills the requirement of a 