figure \[fig:periodic\]. In the exclusion process literature this kind of dynamics is sometimes referred to as discrete time parallel update, see for instance [@Schutz2; @Schutz]. The same structure also appeared in the propagator of the quantum Hirota model [@FaddeevV] and in a deterministic cellular automaton [@MedenjakKP].

iin [0,2,...,6]{} [(i,0) – (i,0.5) ; (i,0.5) – (i,1) ; (i,2) – (i,3) ; (i,4) – (i,5) ; (i,6) – (i,6.5) ; (i,6.5) – (i,7) ;]{} iin [1,2,5,6]{} [(-1,i) – (-0.5,i) ; (-0.5,i) – (0.5,i) ; (1.5,i) – (4.5,i) ; (5.5,i) – (6.5,i) ; (6.5,i) – (7,i) ;]{} iin [3,4]{} [(-0.5,i) – (2.5,i) ; (3.5,i) – (6.5,i) ;]{} iin [0.5,1.5,4.5,5.5]{} [(i,1) – (i,2) ; (i,5) – (i,6) ;]{} iin [-0.5,2.5,3.5,6.5]{} [(i,3) – (i,4) ;]{} at (0,-0.5) \[\] [$1$]{}; at (2,-0.5) \[\] [$2$]{}; at (4,-0.5) \[\] [$3$]{}; at (6,-0.5) \[\] [$4$]{}; at (-1,1.5) \[\] [$U$]{}; at (-1,5.5) \[\] [$U$]{}; at (3,1.5) \[\] [$U$]{}; at (3,5.5) \[\] [$U$]{}; at (7,1.5) \[\] [$U$]{}; at (7,5.5) \[\] [$U$]{}; at (1,3.5) \[\] [$U$]{}; at (5,3.5) \[\] [$U$]{}; at (-2,1.5) \[\] [${{\mathbb U}}^{e}$]{}; at (-2,5.5) \[\] [${{\mathbb U}}^{e}$]{}; at (-2,3.5) \[\] [${{\mathbb U}}^{o}$]{};

The model is said to be integrable because the Markov matrix $M$ commutes with a whole family 