First example is the triple $(X,\pi,B_{0})$ where $X={{\bf R}}^{2n}/\Lambda,\,\,B_{0}={{\bf R}}^{n}/\Lambda^{\prime}$ are tori (here $\Lambda\simeq{{\bf Z}}^{2n},\,\,\Lambda^{\prime}\simeq{{\bf Z}}^{n}$ are lattices), projection $\pi:X\to B_{0}$ is an affine map of tori, and $X$ carries a constant symplectic form. Assuming that fibers of $\pi$ are connected we have $B_{0}=B=B^{sm}$. The monodromy representation is a homomorphism $\rho:\pi_{1}(B)\to{{\bf R}}^{n}\subset GL(n,{{\bf Z}})\ltimes{{\bf R}}^{n}$. Integral affine structure on $B$ depends on $n^{2}$ real parameters, which are coefficients of an invertible $n\times n$ matrix expressing a basis of the lattice $\Lambda^{\prime}\subset T_{x}B$ as a linear combination of generators of the lattice $(T_{x}B)^{{\bf Z}}\subset T_{x}B$, where $x\in B$ is an arbitrary point.

Let $(X,\omega)$ be a surface and $\pi:X\rightarrow B_{0}={\bf R}$ be an arbitrary smooth proper function with isolated critical points. Then $(X,\pi,B_{0})$ is an integrable system. Space $B$ of connected components of fibers is a graph, and ${\bf Z}$-affine structure on $B^{sm}\subset B$ gives a length element on edges of $B$.

Consider a compact connected symplectic manifold $(X,\omega)$ of dimension $2n$ together with a Hamiltonian action of the torus $T^{n}$. Then one has an integrable system $\pi:X\to B_{0}$, where $\pi$ is the moment map of the action and $B_{0}=(Lie(T^{n}))^{\ast}\simeq{{\bf R}}^{n}$. Furthermore, it is well-known that $B=\pi(X)$ is a convex polytope and $B^{sm}$ is the interior of $B$.

Before considering this example let us remark that one can define integrable systems in the case of complex manifolds. More precisely, assume that $X$ is a complex manifold of complex dimension $2n$, $\omega_{\bf C}$ is a holomorphic closed non-degenerate $2$-form on $X$, $B=B_{0}$ is a complex manifold of dimension $n$ and $\pi:X\to B$ is a surjective proper holomorphic map such that generic fibers of $\pi$ are connected complex Lagrangian submanifolds of $X$. With a complex integrable system one can associate a real one by forgetting complex structures on $X$ and $B$ and taking $\omega:=Re(\omega_{\bf C})$ as a symplectic form on $X$. It is easy to see that the image of the monodromy representation belongs to $Sp(2n,{{\bf Z}})\ltimes{{\bf R}}^{2n}\subset GL(2n,{{\bf Z}})\ltimes{{\bf R}}^{2n}$.

Let $(X,\Omega)$ be a complex K3 surface equipped with a non-zero holomorphic 2-form $\omega_{\bf C}=\Omega$ and $\pi:X\to{{\bf C}P}^{1}$ a holomorphic fibration such that the generic fiber of $\pi$ is an elliptic curve. For example, $X$ can be represented as a surface in ${{\bf C}P}^{2}\times{{\bf C}P}^{1}$ given by a general equation $F(x_{0},x_{1},x_{2},y_{0},y_{1})=0$ of bidegree $(3,2)$ in homogeneous coordinates. Map $\pi$ is the projection to the second factor. Holomorphic form $\Omega$ is given by

$$\Omega=i_{Euler_{x}\wedge Euler_{y}}\frac{dx_{0}\wedge dx_{1}\wedge dx_{2}\wedge dy_{0}\wedge dy_{1}}{dF}\,\,,$$

where $Euler_{p}$ denotes the Euler vector field along coordinates $p=(x_{i})$ or $(y_{i})$. Such an elliptic fibration gives an integrable system. Namely, we set $X:=X({{\bf C}})$, $\omega:=Re(\Omega)$, $B:={{\bf C}P}^{1}\simeq S^{2}$. Generically $B^{sing}$ is a set of $24=\chi(X)$ points in $S^{2}$. Singularity of the affine structure near each of $24$ points is well-known in the theory of integrable systems where it is called focus-focus singularity (see e.g. [Au], [Zu]). We will discuss it in Section 6.4. Here we give a short description of this singularity. We take ${{\bf R}}^{2}$ with the standard integral affine structure, remove the point $(x_{0},0)$ on the horizontal axis. Then we modify the affine structure (and also the $C^{\infty}$-structure!) on the ray $\{(x,0)\,|\,\,x>x_{0}\}$. New local integral affine coordinates near points of this ray will be functions $y$ and $x+\max(y,0)$ (see Figure 1). The monodromy of the resulting integral affine structure around removed singular point $(x_{0},0)$ is given by the transformation $(x,y)\mapsto(x+y,y)$.