In the previous example, $\Delta$ was a graph with three edges meeting in one vertex. All three edges were straight lines. In the spirit of Example 3.9 we can perturb each edge of $\Delta$ to a smooth curve starting at the vertex. Each straight edge of the previous example is contained in a $2$-plane which is an integral plane of the distribution spanned by the vectors which are invariant with respect to the holonomy around that edge. For example, consider the edge $E_{1}=\{x_{1}=x_{2}=0,x_{3}\leq 0\}$ of $\Delta$. Then $E_{1}$ is contained inside the half plane, $P_{1}=\{x_{2}=0,x_{3}\leq 0\}$, whose tangent vectors are $T_{1}$ invariant, where $T_{1}=\rho(g_{1})$ is the holonomy of $T_{B_{0}}$ with respect to $E_{1}$. An analogous thing happens with the other two edges. The union of all three half planes gives $R$. The new perturbed edges, $E_{j}^{\prime}$, must be curves inside the half planes $P_{j}$. More precisely, let $\tau$ be a function on $\Delta$ which is the restriction of a smooth function defined on an open neighborhood of $\Delta$, such that $\tau(0)=0$. If we let $R$ be as in the previous example, define

	$\displaystyle\Delta_{\tau}$	$\displaystyle=$	$\displaystyle\{(\tau(q),q)\in\mathbb{R}\times\Delta\}$
	$\displaystyle R^{+}$	$\displaystyle=$	$\displaystyle\{(x_{1},q)\in\mathbb{R}\times\Delta\ |x_{1}\geq\tau(q)\}$
	$\displaystyle R^{-}$	$\displaystyle=$	$\displaystyle\{(x_{1},q)\in\mathbb{R}\times\Delta\ |x_{1}\leq\tau(q)\}$

Now charts $\mathscr{A}=\{U_{i},\phi_{i}\}_{i=1,2}$ on $B-\Delta_{\tau}$ can be defined like in the previous example, but with these new definitions of $R^{+}$ and $R^{-}$. It is clear that $(B,\Delta_{\tau},\mathscr{A})$ defines an affine manifold with singularities. Two different choices of functions $\tau$ define non-isomorphic integral affine manifolds with singularities, unless their graphs inside $R$ can be mapped one to the other via an integral affine map.