  We first formulate an inverse problem for a linear fractional Lam\'e system.
We determine the Lam\'e parameters from exterior partial measurements of the
Dirichlet-to-Neumann map. We further study an inverse obstacle problem as well
as an inverse problem for a nonlinear fractional Lam\'e system. Our arguments
are based on the unique continuation property for the fractional operator as
well as the associated Runge approximation property.
