  Let X be an analytic vector field on a real or complex 2-manifold, and K a
compact set of zeros of X whose fixed point index is not zero. Let A denote the
Lie algebra of analytic vector fields Y on M such that at every point of M the
values of X and [X,Y] are linearly dependent. Then the vector fields in A have
a common zero in K.
  Application: Let G be a connected Lie group having a 1-dimensional normal
subgroup. Then every action of G on M has a fixed point.
