On discretization of Darboux Integrable Systems admitting second-order integrals

We consider a discretization problem for hyperbolic Darboux integrable systems. In particular, we discretize continuous systems admitting $x$- and $y$-integrals of the first and second order. Such continuous systems were classified by Zhyber and Kostrigina. In the present paper, continuous systems are discretized with respect to one of continuous variables and the resulting semi-discrete system is required to be also Darboux integrable.
To obtain such a discretization, we take $x$- or $y$-integrals of a given continuous system and look for a semi-discrete systems admitting the chosen integrals as $n$-integrals. This method was proposed by Habibullin. For all considered systems and corresponding sets of integrals we were able to find such semi-discrete systems. In general, the obtained semi-discrete systems are given in terms of solutions of some first order quasilinear differential systems. For all such first order quasilinear differential systems we find implicit solutions. New examples of semi-discrete Darboux integrable systems are obtained. Also for each of considered continuous systems we determine a corresponding semi-discrete system that gives the original system in the continuum limit.