  We study the fully discrete elliptic integrable model Q4 and its immediate
trigonometric and rational counterparts (Q3, Q2 and Q1). Singular boundary
problems for these equations are systematised in the framework of global
singularity analysis. We introduce a technique to obtain solutions of such
problems, in particular constructing the exact solution on a regular
singularity-bounded strip. The solution technique is based on the
multidimensional consistency and uses new insights into these equations related
to the singularity structure in multidimensions and the identification of an
associated tau-function. The obtained special solutions can be identified with
open boundary problems of the associated Toda-type systems, and have
interesting application to the construction of periodic solutions.
