= Solution
For the <Arnold-Liouville theorem>, work on a <symplectic manifold> of dimension $2d$. Let $F_1=H,F_2,\ldots,F_d$ be <smooth functions> with pairwise vanishing <Poisson brackets>, and suppose
$$
dF_1\wedge\cdots\wedge dF_d\ne0
$$
on a connected component $N$ of a common level. If $N$ is compact, then \b[$N$ is a $d$-dimensional <torus>]. Its commuting <Hamiltonian vector fields> generate translations, and the <Hamiltonian> motion on it is linear in angular coordinates. Compactness ensures completeness of these restricted <vector fields>.
Moreover a neighbourhood of $N$ admits <action-angle variables> $(I_1,\ldots,I_d,\theta_1,\ldots,\theta_d)$, with each angle defined modulo $2\pi$, for which
$$
\{\theta_i,I_j\}=\delta_{ij},\qquad
\{I_i,I_j\}=\{\theta_i,\theta_j\}=0,\qquad
F_j=F_j(I_1,\ldots,I_d).
$$
One compatible <symplectic form> convention is $\omega=\sum_i d\theta_i\wedge dI_i$. The equations become
$$
\boxed{\dot I_i=0,\qquad \dot\theta_i=\frac{\partial H}{\partial I_i},\qquad
\theta_i(t)=\theta_i(0)+t\frac{\partial H}{\partial I_i}(I(0)).}
$$
The resulting motion is periodic when the frequencies are commensurable and otherwise quasiperiodic on a subtorus. This is <Liouville integrability> with $d$ independent <first integrals in involution>. Compactness and regularity cannot be discarded from the <torus> conclusion. On a general <Poisson manifold>, apply this theorem on a regular <symplectic leaf> of dimension $2d$; the ambient dimension alone does not specify the required number of integrals.
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