ABSTRACT
We have shown that the existence of a global first integral for the 2nd order autonomous difference equation is equivalent to finding a set of one parameter Lie group symmetries for which the manifold M is invariant . No discussion has been made about solving the general direct problem of Lie group invariance. That is , given an arbitrary manifold, one wishes to find all admissible Lie group symmetries. Similarly, no discussion has been provided concerning the general inverse problem. That is, given a set of Lie group symmetries, find all manifolds which remain invariant under these groups. In either case, one is basically trying to understand how groups act on manifolds, as well as to construct admissible groups which satisfy certain constraints. These problems however are outside the scope of this paper.
