  We prove that a formal curve $\Gamma$ that is invariant by a $C^\infty$
vector field $\xi$ of $\mathbb{R}^m$ has a geometrical realization, as soon as
the Taylor expansion of $\xi$ is not identically zero along $\Gamma$. This
means that there is a trajectory $\gamma$ of $\xi$ which is asymptotic to
$\Gamma$. This result solves a natural question proposed by Bonckaert nearly
forty years ago. We also construct an invariant $C^0$ manifold $S$ in some open
horn around $\Gamma$ which is composed entirely of trajectories asymptotic to
$\Gamma$, and contains the germ of any such trajectory. If $\xi$ is analytic,
we prove that there exists a trajectory asymptotic to $\Gamma$ which is,
moreover, non-oscillating with respect to subanalytic sets.
