First Integrals/Invariants & Symmetries for Autonomous Difference Equations

For the autonomous kth order difference equation xn+k = f(xn, xn+1, . . . , xn+k−1), where f ∈ C[U ] with domain U ⊂ R, a global first integral/invariant for this difference equation is a nonconstant function H ∈ C[U ], with H : U → R, which remains invariant on the forward orbit Γ(x0, x1, . . . , xk−1). If a first integral exists, then it satisfies a particular functional difference equation and a method of finding solutions is presented. Furthermore, the first integral is constant along the characteristic curves of the associated infinitesimal generator for the Lie group symmetries of the difference equation.