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I am reading this paper: http://aleph.physik.uni-kl.de/~korsch/papers/JPA_24_45.pdf and in section 3 they talk about the integrability of the system. What does that mean?

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The paper refers to the notion of complete integrability, aka. Liouville integrability, i.e. the property that there exist $n$ independent globally defined Poisson-commuting constants of motion. Here $2n$ is the dimension of phase space. In the paper $n=2$. See also this related Phys.SE post.

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