Let's consider the manifold $\mathbb{R}^2$ and the singular distribution $D$ given by $$ D_{(x,y)}=\left\langle\frac{\partial}{\partial x},\varphi(y)\frac{\partial}{\partial y}\right\rangle, \ \ \ \varphi(y)=\left\{\begin{matrix}0&\text{if}&y\leq0,\\ e^{-1/y^2}&\text{if}&y>0.\end{matrix}\right. $$ I know that $D$ is integrable, so, what is the corresponding singular foliation of this distribution? I know that for $y\leq0$ each horizontal line is a leaf, but I am not clear how the foliation is for $y>0$.
Edit: in fact, this is an example due to Sussman that can be found in somewhat more detail in the reference dealing with the problem of integrability of singular distributions.