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Let F be a C∞ vector field defined near the origin O ∈ ℝn, F(O) = 0, and (Ft) be its local flow. Denote by  the set of germs of orbit preserving diffeomorphisms h: ℝn → ℝn at O, and let , (r ≥ 0), be the identity component of  with respect to the weak Whitney Wr topology. Then  contains a subset  consisting of maps of the form Fα(x)(x), where α: ℝn → ℝ runs over the space of all smooth germs at O. It was proved earlier by the author that if F is a linear vector field, then  = .

∞-jets of diffeomorphisms preserving orbits of vector fields