Trajectories of vector fields asymptotic to formal invariant curves

We prove that a formal curve Γ that is invariant by a C∞ vector field ξ of Rᵐ has a geometrical realization, as soon as the Taylor expansion of ξ is not identically zero along Γ . This means that there is a trajectory γ ⊂ Rᵐ of ξ which is asymptotic to Γ . This result solves a natural question proposed by Bonckaert [Smooth invariant curves of singularities of vector fields in R 3 . Ann. Inst. Henri Poincaré 3 (2) (1986), 111–183] nearly forty years ago. We also construct an invariant C⁰ manifold S in some open horn around Γ which is composed entirely of trajectories asymptotic to Γ and contains the germ of any such trajectory. If ξ is analytic, we prove that there exists a trajectory γ asymptotic to Γ which is, moreover, non-oscillating with respect to subanalytic sets.