Darboux-integrable Reductions of the Hirota–Miwa Type Discrete Equations

In the article, the problem of developing a new method for classifying integrable discrete equations with three independent variables of the Hirota–Miwa type is discussed. The classification algorithm is based on the assumption that any three-dimensional integrable lattice admits a hierarchy of reductions in the form of systems of two-dimensional equations of hyperbolic type that are integrable in the sense of Darboux. The results of this work, together with previous works by I.T. Habibullin and A.R. Khakimova on this topic confirm the validity of this assumption at the level of all known examples.