In many examples an integrable system depends on parameters. It often happens that the parameter space ${\cal P}$ carries a natural foliation ${\cal F}$ such that the fundamental group $\pi_{1}({\cal F}_{p},p),p\in{\cal P}$ of any leaf acts on the base space $B_{p}$ of the corresponding torus fibration. This action is given by piecewise-linear homeomorphisms with integral linear parts.

Let us illustrate this phenomenon in the case of the family of integrable systems associated with a K3 surface discussed above.

Here the parameter space ${\cal P}$ has dimension $38$, which is twice of the complex dimension of the space of polynomials $F$ modulo unimodular linear transformations. On the other hand, the miniversal family of representations (up to a conjugation)

$$\left\{\,\rho:\pi_{1}(S^{2}-\{24\mbox{ points}\})\rightarrow SL(2,{\bf Z})\ltimes{\bf R}^{2}\,\right\}$$

such that the monodromy around each puncture is conjugate to $\left(\begin{array}[]{cc}1&1\\ 0&1\end{array}\right)$, has dimension $20$.

Thus, we obtain a foliation ${\cal F}$ of ${\cal P}$ of rank $18=38-20$. It is defined by the following property: if we continuously vary parameters $p\in{\cal P}$ along leaves of ${\cal F}$ then the conjugacy class of the monodromy representation $\rho$ remains unchanged.

Notice that in the local model described above we can move the position $(x_{0},0)$ at which we start the cut. Then we have on the sphere $S^{2}$ a set of $24$ “worms” (singular points, each of them can move in its preferred direction, which is the line invariant under the local monodromy). One can show easily that any continuous deformation of ${\bf Z}$-affine structure satisfying Fixed Point property (see Section 3.1) and preserving the conjugacy class of $\rho$, corresponds to a movement of worms. ²² 2 Notice that in our example $rk({\cal F})=18$ is less than $24$. This means that there are 6 constraints on moving worms.

Moving “worms”we get a canonical identification of manifolds with integral affine structures far enough from singular points. We will see later in Section 6.4 that we also have a canonical PL identification of manifolds near singular points. Therefore we obtain a local system along leaves of ${\cal F}$ with the fiber over $p\in{\cal P}$ being a manifold $B_{p}\simeq S^{2}$ with the above ${\bf Z}$-affine structure. In this way we get a homomorphism from $\pi_{1}({\cal F}_{p},p)$ to $Aut_{{\bf Z}PL}(S^{2})$, where ${\bf Z}PL$ denotes the group of integral PL transformations of $S^{2}$ equipped with the above ${\bf Z}$-affine structure. We will return to this action in Section 6.7 where it will be compared with another PL action on the same space.