  We propose a discrete Darboux-Lax scheme for deriving auto-B\"acklund
transformations and constructing solutions to quad-graph equations that do not
necessarily possess the 3D consistency property. As an illustrative example we
use the Adler-Yamilov type system which is related to the nonlinear
Schr\"odinger (NLS) equation [19]. In particular, we construct an
auto-B\"acklund transformation for this discrete system, its superposition
principle, and we employ them in the construction of the one- and two-soliton
solutions of the Adler-Yamilov system.
