Common zeroes of families of smooth vector fields on surfaces

Let Y and X denote Ck vector fields on a possibly noncompact surface with empty boundary, 1 ≤ k< ∞. Say that YtracksX 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 Poincaré–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 Ck vector fields that track X, then the elements of g have a common zero in K. Applications are made to the dynamics of attractors and transformation groups