Integrable discrete systems

The problem of finding and classifying integrable equations of the form $u^{j+1}_{n,x}=F(u^{j}_{n,x},u^{j+1}_{n},u^{j}_{n+1},u^{j}_{n},u^{j+1}_{n-1})$ with three independent variables is discussed. The presence of two discrete variables poses additional challenges for the study of such models within the known approaches. An efficient algorithm for producing integrable models in this class by an appropriate discretization of the well-understood class of Toda-type lattices with only one discrete variable is proposed. A list containing all previously known integrable models is obtained in the framework of this approach. Lax pairs are provided for all of these equations, some of which were not known before. Bibliography: 45 titles.