We define an affine structure with singularities on $B=\mathbb{R}^{2}$. Let $\Delta=\{0\}$ and let $(x_{1},x_{2})$ be the standard coordinates on $B$. As the covering $\{U_{i}\}$ of $B_{0}=\mathbb{R}^{2}-\Delta$ we take the following two sets

$$U_{1}=\mathbb{R}^{2}-\{x_{2}=0\ \text{and}\ x_{1}\geq 0\},$$

$$U_{2}=\mathbb{R}^{2}-\{x_{2}=0\ \text{and}\ x_{1}\leq 0\}.$$

Denote by $H^{+}$ the set $\{x_{2}>0\}$ and by $H^{-}$ the set $\{x_{2}<0\}$. Let $T$ be the matrix

$$T=\left(\begin{array}[]{cc}1&0\\ 1&1\end{array}\right).$$

(6)

The coordinate maps $\phi_{1}$ and $\phi_{2}$ on $U_{1}$ and $U_{2}$ are defined as follows

	$\displaystyle\phi_{1}$	$\displaystyle=$	$\displaystyle\mathrm{Id}$
	$\displaystyle\phi_{2}$	$\displaystyle=$	$\displaystyle\left\{\begin{array}[]{ll}\mathrm{Id}&\text{on}\ \bar{H}^{+}\cap U_{2},\\ (T^{-1})^{t}&\text{on}\ H^{-}\end{array}\right.$

The atlas $\mathscr{A}=\{U_{i},\phi_{i}\}_{i=1,2}$ is clearly an affine structure on $B_{0}$. It is easy to check that given a point $b\in B_{0}$, we can chose a basis of $T^{\ast}_{b}B_{0}$ with respect to which the holonomy representation $\rho^{\ast}$ sends the anti-clockwise oriented generator of $\pi_{1}(B_{0})$ to the matrix $T$.