On constructing solutions for the functional equation Z(x,y,n)=Z(a11x+a12y,a21x+a22y,n+1)

The concept of invariance factors to obtain first integrals for difference equations was presented by Van Horssen (2002) [6]. It was shown that all invariance factors have to satisfy a functional equation. One of the main difficulties in finding first integrals for a system of first order difference equations is solving the aforementioned functional equation. In this paper, we consider a functional equation which is related to a system of two first order, linear difference equations. Linear transformations and an adapted version of the method of separation of variables will be used to construct solutions of this functional equation. The constructed solutions of this functional equation do not seem to be available in the literature.