= Solution
The <Arnold-Liouville theorem> concerns a connected compact component $L$ of a regular common level of $n$ commuting independent smooth functions $F=(F_1,\ldots,F_n)$ on a $2n$-dimensional <symplectic manifold>. It states that $L$ is a <Lagrangian torus>, and that a neighbourhood of $L$ has <action-angle variables>
$$
(\theta,I)\in\mathbb T^n\times U,\qquad
\omega=\sum_jd\theta_j\wedge dI_j,
$$
in which every $F_i$ depends only on $I$. If $H=F_1$ or, more generally, $H=h(F)$, its <Hamiltonian flow> satisfies
$$
\boxed{\dot I=0,\qquad\dot\theta=\nabla_I H(I).}
$$
Angles have period $2\pi$ here. The theorem is local near this regular compact component; it does not assert globally defined <action-angle variables> across singular levels or over an entire base with monodromy.
First, $L$ has <dimension of a manifold> $n$ by the <submersion theorem>. The <Hamiltonian vector fields> $X_{F_i}$ are independent, tangent to $L$, and commute. They span $TL$. Moreover
$$
\omega(X_{F_i},X_{F_j})=dF_i(X_{F_j})=\{F_i,F_j\}=0,
$$
so $L$ is a <Lagrangian submanifold>. Compactness makes these restricted <vector fields> complete. Their joint flow is an $\mathbb R^n$-action on $L$. Its orbits are open because the fields span $TL$, so connectedness gives one orbit. The stabilizer $\Lambda$ of a point is discrete by the <inverse function theorem>, and
$$
L\cong\mathbb R^n/\Lambda.
$$
Compactness forces $\Lambda$ to be a full-rank <Euclidean lattice>: otherwise an unbounded linear coordinate transverse to its span would descend to the quotient. Thus \b[$L$ is an $n$-torus].
Next choose a small ball of regular values near $F(L)$ and a neighbourhood $N$ of this component which is a product family of compact tori $L_c$. This local trivialization follows directly by choosing transverse <vector fields> $Z_j$ with $dF_i(Z_j)=\delta_{ij}$ and lifting short radial paths in the base; compactness of $L$ gives a uniform neighbourhood where their flows exist. Hence $N$ deformation retracts onto $L$. The commuting joint flows on nearby fibres have smoothly varying full-rank <Hamiltonian period lattices>. A basis $t^{(j)}(c)\in\mathbb R^n$ of their periods can be chosen smoothly on this small ball: continue the return equations from a basis at the central fibre, using their nonsingular vertical flow derivatives and the <implicit function theorem>. No global choice of lattice basis is needed.
Because $\omega|_L=0$ and $N$ retracts onto $L$, its closed <symplectic form> is exact on $N$. Choose a one-form $\lambda$ with $d\lambda=-\omega$. Let $\gamma_j(c)$ be the smoothly continued cycles represented by the period basis and define the <action integrals>
$$
I_j(c)=\frac1{2\pi}\int_{\gamma_j(c)}\lambda.
$$
For a transverse variation $v$ of the fibre, differentiating a cycle integral and using <Cartan's magic formula> eliminates the integral of the exact term. With the cycle parametrized by the joint-flow time $t^{(j)}$, this gives
$$
dI_j(v)=\frac1{2\pi}\int_{\gamma_j}(-\omega(v,\dot\gamma_j))
=\frac1{2\pi}\sum_i t_i^{(j)}(c)\,dF_i(v).
$$
Thus
$$
\boxed{dI_j=\frac1{2\pi}\sum_i t_i^{(j)}(c)\,dc_i.}
$$
The period matrix is nonsingular, so $I$ gives coordinates on the base. The corresponding <Hamiltonian vector fields> satisfy
$$
X_{I_j}=\frac1{2\pi}\sum_i t_i^{(j)}(c)X_{F_i}.
$$
Their time-$2\pi$ flows return along the basis cycles. They commute because each $I_j$ is a function of the commuting $F_i$. Consequently they give a free <torus action> on $N$.
Choose a section over the action-coordinate ball and use this <torus action> to define angles $\theta$. Since $X_{I_j}=\partial_{\theta_j}$, the <symplectic form> takes the form
$$
\omega=\sum_jd\theta_j\wedge dI_j+\beta(I).
$$
The closed base two-form $\beta$ is exact on the ball: write $\beta=d\alpha$, $\alpha=\sum_j a_j(I)dI_j$, by the <Poincaré lemma>. Changing the angular origins by $\theta'_j=\theta_j+a_j(I)$ removes this term, since $\sum_jd\theta'_j\wedge dI_j=\omega$. This completes the construction of <action-angle variables>. The functions $F_i$ are constant on the fibres, so are functions of $I$ alone, and <Hamilton's equations> give the claimed straight-line motion. This proves both the topological and symplectic parts of the <Arnold-Liouville theorem>.
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