  Let $G$ be connected nilpotent Lie group acting locally on a real surface
$M$. Let $\varphi$ be the local flow on $M$ induced by a $1$-parameter
subgroup. Assume $K$ is a compact set of fixed points of $\varphi$ and $U$ is a
neighborhood of $K$ containing no other fixed points.
  Theorem: If the Dold fixed-point index of $\varphi_t|U$ is nonzero for
sufficiently small $t>0$, then ${\rm Fix} (G) \cap K \ne \emptyset$.
