Consider the discrete dynamical system generated a map F. It is said that it is globally periodic if there exists a natural number p such that F p (x) ≡ x. On the other hand it is called completely integrable if it has as many functionally independent first integrals as the dimension of the phase space. In this paper we relate both concepts. We also give a large list of globally periodic dynamical systems together with a complete set of their first integrals, emphasizing in the ones coming from

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... difference equations. 2000 Mathematics Subject Classification. 39A11, 39A20. Key words and phrases. Globally periodic discrete dynamical system, first integrals, completely integrable systems, difference equations, invariants for difference equations.