  Let Y and X denote C^k vector fields on a possibly noncompact surface with
empty boundary, k >0. Say that Y tracks X if the dynamical system it generates
locally permutes integral curves of X. Let K be a locally maximal compact set
of zeroes of X.
  THEOREM Assume the Poincar'e-Hopf index of X at K is nonzero, and the k-jet
of X at each point of K is nontrivial. If g is a supersolvable Lie algebra of
C^k vector fields that track X, then the elements of g have a common zero in K.
  Applications are made to attractors and transformation groups.
