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