Assume that we are given a non-compact manifold $B^{sm}$ with a ${\bf Z}$-affine structure. We would like to “compactify” it, i.e. to find a compact Hausdorff topological space $B$ such that $B^{sm}\subset B$ is an open dense subset. We do not require an extension of the ${\bf Z}$-affine structure to $B$. The question is: what kind of properties one should expect from such a compactification? We cannot give a complete list of such properties at the moment. Instead, we formulate two of them and illustrate the notion of compactification in PL case. Similarity between examples in Sections 3.2 and 4.2 suggests that the class of singularities which appear in integrable systems should be more or less the same as the class of singularities appearing in non-archimedean geometry.

Let $x\in B^{sing}:=B\setminus B^{sm}$ be a singular point of some compactification of $B^{sm}$. Then we require the following
Finiteness property. There is a fundamental system of neighborhoods $U\subset B$ of $x$ such that the number of connected components of $U\cap B^{sm}$ is finite.

Let $U\cap B^{sm}=\sqcup_{1\leq i\leq N}U_{i}$ be the disjoint union of the connected components. Let us pick a point $x_{i}\in U_{i}$ and consider a continuous path $\gamma:[0,1]\to B$ such that $\gamma(0)=x_{i},\gamma(1)=x,\gamma([0,1))\subset U_{i}$. Using the affine structure we can canonically lift this path to a path $\gamma^{\prime}:[0,1)\to T_{x_{i}}B$. We assume that the lifted path $\gamma^{\prime}$ extends to time $t=1$ and is analytic at $t=1$ (it is a technical assumption, helping to avoid some pathologies). Then we require the following
Independence property. Path $\gamma$ with the properties as above exists, and point $\gamma^{\prime}(1)\in T_{x_{i}}B$ does not depend on the choice of $\gamma$.

Independence property implies the existence of a fixed vector for the monodromy representation restricted to $\pi_{1}(U\cap B^{sm},x_{i})$ (this implies the Fixed Point property from Section 3.1).