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.



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> What is an example of such singular foliation for which the foliation is stable at $0$?
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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](http://qcpages.qc.cuny.edu/~zakeri/papers/SHFC_zak.pdf).



Stability of singularity in singular holomorphic foliation
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).


