In the previous example the discriminant locus $\Delta$ was a straight line. We can slightly perturb $\Delta$ so that it becomes a smooth curve. More precisely, let $B=\mathbb{R}^{2}\times I$ as before and consider a smooth function $\tau:I\rightarrow\mathbb{R}$. Let

$$\Delta_{\tau}=\{(\tau(s),0,s),s\in I\}\subset B$$

and define a covering $\{U_{i}\}$ of $B_{0}=B-\Delta_{\tau}$ to be

$$U_{1}=(\mathbb{R}^{2}\times I)-\{(x_{1},0,s)\ |\ x_{1}\geq\tau(s)\},$$

$$U_{2}=(\mathbb{R}^{2}\times I)-\{(x_{1},0,s)\ |\ x_{1}\leq\tau(s)\}.$$

Now let $H^{+}=\{x_{2}>0\}$ and $H^{-}=\{x_{2}<0\}$. Take the following matrix

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

and define maps $\phi_{j}$ on $U_{j}$ to be

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

Clearly $\mathscr{A}=\{U_{i},\phi_{i}\}_{i=1,2}$ defines an affine structure on $B_{0}=B-\Delta_{\tau}$. When $\tau=0$, this example coincides with the previous one. Notice that the curve $(\tau(s),0,s)$ is contained inside the $2$-plane $\{x_{2}=0\}$, which can be viewed as an integral surface of the distribution spanned by the vectors in $TB_{0}$ which are invariant with respect to the holonomy representation $\rho$ on $TB_{0}$. Two different curves give non-isomorphic singular affine structures, unless the curves can be taken one into the other via an integral affine transformation.