  The first part of this paper is concerned with the uniqueness to inverse
time-harmonic elastic scattering from bounded rigid obstacles in two
dimensions. It is proved that a connected polygonal obstacle can be uniquely
identified by the far-field pattern over all observation directions
corresponding to a single incident plane wave. Our approach is based on a new
reflection principle for the first boundary value problem of the Navier
equation. In the second part, we propose a revisited factorization method to
recover a rigid elastic body with a single far-field pattern.
