© Cambridge University Press, 2016. 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 @M. The action need not be smooth. Let φ 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 φ and U is a neighborhood of K containing no other fixed points. THEOREM. If the Dold fixed-point index of φt|U is non-zero for sufficiently small t > 0, then Fix(G) ∩ K ≠ Ø.