# Why and how almost periodic series constitute the algebra of observable of integrable systems?

In the introduction of his book Noncommutative Geometry, p. 42, Connes explains that when a classical dynamical system has enough constants of motions, the motion of the system is almost periodic, and the algebra of observables is the algebra of almost periodic series

$$q(t) = \sum q_{n_1 \dots n_k} \, \exp( 2 \pi i \left\langle n, \nu \right\rangle t),$$

where $$n_i$$ are integers and $$\nu_i$$ "fundamental frequencies" (and $$\left\langle n, \nu \right\rangle = \sum n_i \nu_i$$).

From what does this come from?

I believe that "enough constants of motion" means that the system is somehow integrable, and perhaps the Arnold-Liouville theorem can help provide an answer to this question, but I am not familiar with this domain, and I hope someone will know better what Cones means.