We prove that for each integer $k\ge 2$ there is an open neighborhood ${𝒱}_k$ of the identity map of the 2-sphere $S^2$, in $C^1$ topology such that: if $G$ is a nilpotent subgroup of ${\mathrm{Diff}}^1\left(S^2\right)$ with length $k$ of nilpotency, generated by elements in ${𝒱}_k$, then the natural $G$-action on $S^2$ has nonempty fixed point set. Moreover, the $G$-action has at least two fixed points if the action has a finite nontrivial orbit.

We describe necessary and sufficient conditions for a fixed point free planar homeomorphism that preserves the standard Reeb foliation to embed in a planar flow that leaves the foliation invariant.

We consider a fixed point free homeomorphism $h$ of the closed band $B=ℝ\times [0,1]$ which leaves each leaf of a Reeb foliation on $B$ invariant. Assuming $h$ is the time one of various topological flows, we compare the restriction of the flows on the boundary.