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Assume M is a 3-dimensional real manifold without boundary, A is an abelianLie algebra of analytic vector fields on M, and X is an element of A. Thefollowing result is proved:  If K is a locally maximal compact set of zeroes of X and the Poincar'e-Hopfindex of X at K is nonzero, there is a point in K at which all the elements ofA vanish.

Zero sets of Abelian Lie algebras of vector fields