Symmetries and first integrals of ordinary difference equations

This paper describes a new symmetry-based approach to solving a given ordinary difference equation. By studying the local structure of the set of solutions, we derive a systematic method for determining one-parameter Lie groups of symmetries in closed form. Such groups can be used to achieve successive reductions of order. If there are enough symmetries, the difference equation can be completely solved. Several examples are used to illustrate the technique for transitive and intransitive symmetry groups. It is also shown that every linear second-order ordinary difference equation has a Lie algebra of symmetry generators that is isomorphic to sl(3). The paper concludes with a systematic method for constructing first integrals directly, which can be used even if no symmetries are known.

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