Let $(X,\omega)$ be a smooth symplectic manifold of dimension $2n$, $B_{0}$ a smooth manifold of dimension $n$, $\pi:X\to B_{0}$ a smooth map with compact fibers, such that $\{\pi^{\ast}(f),\pi^{\ast}(g)\}=0$ for any $f,g\in C^{\infty}(B_{0})$. Here $\{\cdot,\cdot\}$ denotes the Poisson bracket on $X$. We assume that $\pi$ is a submersion on an open dense subset $X^{\prime}\subset X$. Such a triple $(X,\pi,B_{0})$ is called an integrable system. In applications it is typically given by a collection of smooth functions $(H_{1},...,H_{n})$ on $X$ (these functions are called Hamiltonians) such that $\{H_{i},H_{j}\}=0,1\leq i,j\leq n$. Usually first Hamiltonian $H=H_{1}$ is identified with the energy of mechanical system.

Let us consider the case when $\pi$ is proper. It is a natural restriction, because in applications the energy $H_{1}$ is already a proper map $H_{1}:X\to{{\bf R}}$.

Let $x\in B_{0}$ be a point such that the restriction of $\pi$ to $\pi^{-1}(x)$ is a submersion. We call such points $\pi$-smooth. According to Sard theorem $\pi$-smooth points form an open dense subset of $B_{0}$. The fiber $\pi^{-1}(x)$ is a compact Lagrangian submanifold of $X$. The Liouville integrability theorem (see [Ar]) says that $\pi^{-1}(x)$ is a disjoint union of finitely many tori $T^{n}_{\alpha}$. Moreover, for each torus $T^{n}_{\alpha}$ there exists a local coordinate system $(\varphi_{1},...,\varphi_{n},I_{1},...,I_{n})$ in a neighborhood $W_{\alpha}$ of $T^{n}_{\alpha}$ such that $\varphi_{i}\in{{\bf R}}/2\pi{{\bf Z}},(I_{1},...,I_{n})\in{{\bf R}}^{n}$ and $\omega=\sum_{1\leq i\leq n}dI_{i}\wedge d\varphi_{i}$. These coordinates are called action-angle coordinates. The map $\pi$ in action-angle coordinates is given by the projection $(\varphi_{1},...,\varphi_{n},I_{1},...,I_{n})\mapsto(I_{1},...,I_{n})$. There is an ambiguity in the choice of action-angle coordinates. In particular action coordinates $I=(I_{1},...,I_{n})$ are defined up to a transformation $I^{\prime}=A(I)+b,A\in GL(n,{{\bf Z}}),b\in{{\bf R}}^{n}$. Indeed, the free abelian group generated by $1$-forms $dI_{i},1\leq i\leq n$ in each cotangent space $T_{x}^{\ast}B_{0}$ admits an invariant description. It is the free abelian group generated by the restrictions of $1$-forms $\int_{\gamma}\omega$ to $T_{x}^{\ast}B_{0}$, where $\gamma$ runs through closed singular $1$-chains in $\pi^{-1}(x)\cap W_{\alpha}$. In this way we obtain a ${\bf Z}$-affine structure on $\pi(W_{\alpha})$.

Let $B$ be the set of connected components of fibers of $\pi$. Endowed with the natural topology it becomes a locally compact Hausdorff space, projection from $X$ to $B$ will be denoted by the same letter $\pi$. The natural continuous map $B\to B_{0}$ is a kind of “ramified finite covering”. Let us define $B^{sm}\subset B$ as the set of connected components on which $\pi$ is a submersion (i.e. the set of all Liouville tori). Then $B^{sm}$ is an open dense subset in $B$. Hence it carries a ${\bf Z}$-affine structure given by the action coordinates.

The singular part $B^{sing}=B\setminus B^{sm}$ consists of projections of singular fibers. Typically the codimension of $B^{sing}$ is greater or equal to $1$. The codimension $1$ stratum consists of the boundary of the image of $\pi$ and of the ramification locus of the map $B\to B_{0}$. The structure of singularities of the integral affine structure in higher codimensions is less understood. It seems that the following property is always satisfied:
Fixed Point property . For any $x\in B^{sing}$ there is a small neighborhood $U$ such that the monodromy representation $\pi_{1}((U\setminus B^{sing})_{\alpha})\to GL(n,{{\bf Z}})\ltimes{{\bf R}}^{n}$ for any connected component $(U\setminus B^{sing})_{\alpha}$ of $U\setminus B^{sing}$ has a fixed vector in ${{\bf R}}^{n}$ in the natural representation by affine transformations.

We will discuss this property in Section 6 devoted to compactifications.