Stability of singularity in singular holomorphic foliation For  an open subset $U$ of $\mathbb{C}^{2}$  containing $0$ and  a  holomorphic  map $f:U\to \mathbb{C}^{2}$  which has a unique zero at the origin we  associate a natural singular holomorphic foliation  by complex curves, the foliation arising from $\dot z=f(z)$. In this case origin is  called a singularity.

What is an example of such singular foliation for which the foliation is stable at $0$?

That is,  for every open set $W$ containing origin, there is an open set $V\subset W$ which saturation under foliation is contained in $W$? Is there an indirect relation between this question and  some methods  in "Minimal set problem" about singular holomorphic foliations of $\mathbb{C}P^{2}$?
Note that a  minimal set for  a  singular foliation of  $\mathbb{C}P^{2}$ is a  compact subset which is invariant(saturated) under foliation and does not contain any singular point. For "minimal set" see here.
 A: Such an example is impossible. We can always assume $W$ is a polydisc, and part of its boundary $\partial W$ is included in the $3$-space $T=\{(x,y) : |y|=r\}$. Take $p\in T\cap\bar W$. If the foliation were stable then the image of $t\mapsto z(t))$,  for small $t$ and with $z(0)=p$, would be included in the adherence of a single connected component of $\mathbb C^2\setminus T$. This contradicts the maximum/minimum modulus  principle for the $y$-component of $z(t)$ except if $z=cst$, but in that case the trajectory escapes from $W$ through another component of the boundary. Therefore every leaf of the foliation passing through $\partial W$ escapes from $W$. 
You can adapt this construction to the saturation $S$ of $V$ in $W$, to prove the foliation always reaches $\partial W$. If it were not the case,  consider the smallest closed polydisc $\bar W_S\subsetneq \bar W$ containing $S$. There is a point $p\in\partial S \cap \partial W_S$, to which the above argument applies: either the trajectory passing through $p$ escapes from $W_S$ (contradiction) or it is included in $\partial W_S$ and therefore leaves the polydisc through another component of $\partial W_S$ (contradiction again).
