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.