Again, we can perturb the above example by replacing the straight edges of $\Delta$ with smooth curves starting at the origin. This time these curves have to be contained inside $\{0\}\times\mathbb{R}^{2}$, which is the integral surface (containing $\Delta$) of the distribution spanned by the $\rho$-holonomy invariant vectors in $TB_{0}$. The perturbed $\Delta$, which we could denote $\Delta_{\tau}$, still separates $\mathbb{R}^{2}$ in three connected components $C_{1},C_{2}$ and $C_{3}$. Then the definition of the affine structure carries through just like in the previous example and we denote it by $\mathscr{A}_{\tau}$.