Zero sets of Abelian Lie algebras of vector fields
  Assume M is a 3-dimensional real manifold without boundary, A is an abelian Lie algebra of analytic vector fields on M, and X is an element of A. The following result is proved:   If K is a locally maximal compact set of zeroes of X and the Poincar'e-Hopf index of X at K is nonzero, there is a point in K at which all the elements of A vanish. 
Morris W. Hirsch
math.DS