which components of the tensor space the operators are acting non-trivially. For instance $K_1(z) = K(z) \otimes 1$ and $K_2(z) = 1 \otimes K(z)$. The matrix $\overline{K}(z)$ is associated to the right boundary and satisfies the reversed reflection equation $$\label{eq:reflection_equation_reversed}
  R_{12}\left(\frac{z_1}{z_2}\right)^{-1} \overline K_1(z_1) R_{21}(z_1 z_2)^{-1} \overline K_2(z_2) =
  \overline K_2(z_2) R_{12}(z_1 z_2)^{-1} \overline K_1(z_1) R_{21}\left(\frac{z_1}{z_2}\right)^{-1}.$$

We require also that the $K$-matrices satisfy further properties. We recall that we are interested in constructing discrete time Markov matrices. We therefore impose that the $K$-matrices fulfill the Markovian property $$\label{eq:markovian_K}
 {{\langle\sigma|}}K(z)={{\langle\sigma|}} \quad \mbox{and} \quad {{\langle\sigma|}}\overline{K}(z)={{\langle\sigma|}}$$ which is nothing else than demanding that the sum of the entries on each column of the boundary matrices is equal to one. In our construction the $K$-matrices will indeed play the role of local Markovian operators on the boundaries. We also require the boundary matrices to satisfy the regularity property $$\label{eq:regularity_K}
 K(1) = 1 \quad \mbox{and} \quad \overline{K}(1) = 1$$ and the unitarity property $$\label{eq:unitarity_K}
 K(z)K\left(\frac{1}{z}\right) = 1 \quad \mbox{and} \quad \overline{K}(z)\overline{K}\left(\frac{1}{z}\right) = 1.$$

Once again we wrote all the definitions and properties about the $K$-matrices assuming that the spectral parameter is . We can easily translate them to an spectral parameter (we give them for the 