LetKbe a fieldandsets of indetorminates, andf∈k[[x]]g∈k[[Y]] two non-zero formal power series withf∈ (x)g∈ (Y). Setand suppose that δgis a (y)-primary ideal (e.g., if charK= 0 and g is an isolated singularity). WriteR=K[[X,Y]]/(f+g)R1=k[[X]]/(f) and [Rtilde]:=R/δgR. The main aim of this paper is to relate an arbitrary maximal Cohen-Macaulay (MCM for short)R-moduleNto the higher order syzygy, and in this way relate indecomposable MCMR-modules to higher order syzygies of certain indecomposable[Rtilde]-modules. The[Rtilde]-modules in question are deformations of MCMR-modules and are weakly liftablc. We find resolutions of the higher order syzygy modules in question which are shown to be minimal in certain situations, and express these in terms of matrix factorizations. The theory is shown to be applicable with almost complete success to singularities of KnOrrer-type and, in any case, to give detailed information about MCMR-modules.