Abstract
In this paper we present a review of results on the integrability of dynamical systems and its detection. Three different approaches are examined. The first introduces the Kowalevski exponents as related to the singularity structure of some weighted homogeneous differential systems. It is proved that whenever these exponents are irrational or complex, the system cannot be integrable. The second approach is known as the ‘weak Painleve’ criterion. No (precise) general theorem can be given in this case. However, in the vast majority of cases, irrational or complex Painleve resonances (which can be different from the Kowalevski exponents in some cases) are indications of nonintegrability. Finally, the method of Ziglin is presented, along with an extension. Ziglin's theorem deals with the branching of the solutions and uses nonlocal information obtained through the monodromy properties around solutions of a special type in order to establish the necessary conditions for integrability.
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