Let $(M, H)$ be a completely integrable $2n$-dimensional Hamiltonian system with first integrals $f_1, \ldots, f_n$. For a fixed value $c \in \mathbb{R}^n$, define the level set \begin{align*} M_c = \{(q, p) \in M : f_i(q, p) = c_i,\; i = 1, \ldots, n\}. \end{align*} Then: 1. $M_c$ is a smooth submanifold of $M$ that is invariant under the Hamiltonian flow. If $M_c$ is compact and connected, then it is diffeomorphic to the $n$-torus $T^n = S^1 \times \cdots \times S^1$. 2. If $M_c$ is compact and connected, there exist canonical coordinates $(\phi, I)$ — the **action-angle variables** — defined locally near $M_c$, in which the equations of motion take the form \begin{align*} \dot{I} = 0, \quad \dot{\phi} = \frac{\partial \widetilde{H}}{\partial I} = \text{constant}. \end{align*}