Fixed points of local actions of nilpotent Lie groups on surfaces

Let G be a connected nilpotent Lie group with a continuous local action on a real surface M, which might be non-compact or have non-empty boundary partial derivative M. The action need not be smooth. Let phi be the local flow on M induced by the action of some one-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)backslash U is non-zero for sufficiently small t > 0, then Fix(G) boolean AND K not equal empty set.