Consider an $n$ degree of freedom Hamiltonian which admits $m$ first integrals and, when restricted to a specific level of these first integrals, then admits $n-m$ first integrals. We will see that this notion of integrability does not coincide with Liouville Arnold integrability. Still, most of the classical results hold: for most systems, the dynamic is still quasi periodic on a full measure set of the phase space, and the system is still integrable by quadratures. However, there is also a dense set on which the motion is not quasi periodic, and the angle coordinates are not smooth functions of the actions. We will also explore the consequences for the reduction of Hamiltonian admitting first integrals.
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