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 $(x,y)\mapsto(x+y,y)$.