Algebraic reductions of discrete equations of Hirota-Miwa type

For nonlinear discrete equations in the dimension $1+1$ there are easily checked symmetry criterions of integrability which lie in the base of the classification algorithms. A topical problem on creating effective methods for classifying integrable discrete equations with three or more independent variables remains open, since in the multidimensional case the symmetry approach loses its effectiveness due to difficulties related with non-localities.
In our recent works we discovered a specific property of discrete equations in the three-dimensional case which seems to be an effective criterion for the integrability of three-dimensional equations. It turned out that many known integrable chains including equations like two-dimensional Toda chain, equation of Toda type with one continuous and two discrete independent variables, equations of Hirota-Miwa type, where all independent variables are discrete are characterized by the fact that they admit cut-off conditions of special form in one of discrete variables which reduce the chain to a system of equations with two independent variables possessing an increased integrability; they possess complete sets of the integrals in each of the characteristics, that is, they are integrable in the Darboux sense. In other words, the characteristic algebras of the obtained finite-field systems have a finite dimension. In this paper, we give examples confirming the conjecture that the presence of a hierarchy of two-dimensional reductions integrable in the Darboux sense is inherent in all integrable discrete equations of the Hirota-Miwa type. Namely we check that the lattice Toda equation and its modified analogue also admit the aforementioned reduction.