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We propose a discrete Darboux–Lax scheme for deriving auto-Bäcklund 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 Schrödinger (NLS) equation [7]. In particular, we construct an auto-Bäcklund 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. 02.30.Ik, 02.90.+p, 03.65.Fd 37K60, 39A36, 35Q55, 16T25. • A new method for solving integrable nonlinear partial diference equations (P Δ Es) • Systematic derivation of soliton solutions to integrable nonlinear P Δ Es • Study of an Adler-Yamilov type of system which discretises the NLS equation • Construction of Darboux and Bäcklund transformations for the Adler-Yamilov system • Derivation of one - and two-soliton solutions of the Adler-Yamilov system [ABSTRACT FROM AUTHOR]