  This paper concerns the time-harmonic direct and inverse elastic scattering
by an extended rigid elastic body surrounded by a finite number of point-like
obstacles. We first justify the point-interaction model for the Lam\'{e}
operator within the singular perturbation approach. For a general family of
pointwise-supported singular perturbations, including anisotropic and non-local
interactions, we derive an explicit representation of the scattered field. In
the case of isotropic and local point-interactions, our result is consistent
with the ones previously obtained by Foldy's formal method as well as by the
renormalization technique.
  In the case of multiple scattering with pointwise and extended obstacles, we
show that the scattered field consists of two parts: one is due to the
diffusion by the extended scatterer and the other one is a linear combination
of the interactions between the point-like obstacles and the interaction
between the point-like obstacles with the extended one.
  As to the inverse problem, the factorization method by Kirsch is adapted to
recover simultaneously the shape of an extended elastic body and the location
of point-like scatterers in the case of isotropic and local interactions. The
inverse problems using only one type of elastic waves (i.e. pressure or shear
waves) are also investigated and numerical examples are present to confirm the
inversion schemes.
