Let us remove the angle $\left\{(x,y)\in{\bf R}^{2}\,|\,\,0<x<y\,\right\}$ from ${{\bf R}}^{2}$. After that we identify sides of the angle by the affine transformation $(x,y)\mapsto(x+y,y)$. In this way we introduce a new ${\bf Z}$-affine structure on ${{\bf R}}^{2}\setminus\{(0,0)\}$ with the monodromy around $(0,0)$ given by the unipotent matrix (see Figure 3)

This ${\bf Z}$-affine structure does not admit a continuation to ${{\bf R}}^{2}$. Therefore we obtain a ${\bf Z}$-affine structure with singularities on ${{\bf R}}^{2}$. We will call standard the singularity at $(0,0)$.

Equivalently, we can describe this ${\bf Z}$-affine structure on ${{\bf R}}^{2}\setminus\{(0,0)\}$ by taking a cut along the ray $\{(x,0)\,|\,x>0\}$ in ${{\bf R}}^{2}$ and glue the standard ${\bf Z}$-affine structure above and below the cut by means of the affine transformation $(x,y)\mapsto(x+y,y)$ (see Figure 1 in Section 3.2.4). In this description it is clear that we can start the cut at arbitrary point $(x_{0},0)$ on the $x$-axes. The resulting singularity will be also called the standard one.

Standard singularity is called focus-focus singularity in the theory of integrable systems (see [Zu]). In non-archimedean geometry it appears as a singular value of some map $f:X^{an}\to{{\bf R}}^{2}$, where $X$ is an algebraic surface in $3$-dimensional affine space ${\bf A}_{K}^{3}$ (see Section 8).

Let us consider the Cartesian product of ${{\bf R}}^{2}\setminus\{(0,0)\}$ equipped with the above ${\bf Z}$-affine structure with the standard (non-singular) ${\bf Z}$-affine structure on ${{\bf R}}^{n-2}$. Let us choose a continuous function $f(z_{1},...,z_{n-2})$ and start the cuts at all points $(f(z_{1},...,z_{n-2}),0,z_{1},...,z_{n-2})$. This means that we introduce the standard non-singular ${\bf Z}$-affine structure in the region $y\neq 0$ as well as in the region $(y=0,x<f(z_{1},...,z_{n-2}))$. Near points $(y=0,x>f(z_{1},...,z_{n-2}))$ we introduce a modified ${\bf Z}$-affine structure by declaring functions

to be ${\bf Z}$-affine coordinates. This gives an example of a “curved” singular set $B^{sing}$ of codimension 2. Since function $f$ can be approximated by PL functions, the above ${\bf Z}$-affine structure can be deformed to a PL one.