We study the inverse problem in difference Galois theory of linear differential equations over difference-differential field C(x) with derivation ddx and endomorphism f(x)?f(x+1). Our main result is that every algebraic group, considered as a occurs group some equation C(x).

We establish constructive conditions for the solvability of linear boundary-value problem a system difference-algebraic equations and develop scheme construction solutions this problem. also obtain sufficient reducibility to sequence systems combining difference algebraic equations. propose an original classification unified problems.