Fixed points of local actions of nilpotent Lie groups on surfaces

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