= Solution
An autonomous <Hamiltonian> system with $n$ degrees of freedom is <Liouville integrable> if it has $n$ conserved functions $I_1,\ldots,I_n$, including the <Hamiltonian>, whose differentials are independent on a regular open set and which are in involution:
$$
\boxed{dI_1\wedge\cdots\wedge dI_n\ne0,\qquad\{I_i,I_j\}=0.}
$$
The equations can then be reduced locally to quadratures. The statement about action-angle coordinates on invariant tori additionally requires compact connected regular common level sets; integrable unbounded motion need not be periodic or lie on a torus.
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