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Kovalevskaya, Liapounov, Painlevé, Ziglin and the Differential Galois Theory

We give a review about the integrability of complex analytical dynamical systems started with the works of Kovalevskaya, Liapounov and Painleve as well as by Picard and Vessiot at the end of the XIX century. In particular, we state a new result which generalize a theorem of Ramis and the author. This last theorem is itself a generalization of Ziglin’s non-integrability theorem about the monodromy group of the first order variational equation. Also we try to point out some ideas about the connection of the above results with the Painleve property.