  The article considers lattices of the two-dimensional Toda type, which can be
interpreted as dressing chains for spatially two-dimensional generalizations of
equations of the class of nonlinear Schr\"odinger equations. The well-known
example of this kind of generalization is the Davey-Stewartson equation. It
turns out that the finite-field reductions of these lattices, obtained by
imposing cutoff boundary conditions of an appropriate type, are Darboux
integrable, i.e., they have complete sets of characteristic integrals. An
algorithm for constructing complete sets of characteristic integrals of finite
field systems using Lax pairs and Miura-type transformations is discussed.
