We consider second-order quasilinear elliptic systems on un-bounded domains in the setting of Sobolev spaces. We complete our earlier work on the Fredholm and properness properties of the associated differential operators by giving verifiable conditions for the linearization to be Fredholm of index zero. This opens the way to using the degree for C-1-Fredholm maps of index zero as a tool in the study of such quasilinear systems. Our work also enables us to check the Fredholm assumption which plays an important role in Rabies approach to proving exponential decay to zero at infinity of solutions. (C) 2010 Elsevier Inc. All rights reserved.