  We study certain significant properties of the equilibrium configurations of
a rigid body subject to an undamped elastic restoring force, in the stream of a
viscous liquid in an unbounded 3D domain. The motion of the coupled system is
driven by a uniform flow at spatial infinity, with constant dimensionless
velocity $\lambda$. We show that if $\lambda$ is below a critical value,
$\lambda_c$ (say), there is a unique and stable time-independent configuration,
where the body is in equilibrium and the flow is steady. We also prove that, if
$\lambda<\lambda_c$, no oscillatory flow may occur. Successively, we
investigate possible loss of uniqueness by providing necessary and sufficient
conditions for the occurrence of a steady bifurcation at some $\lambda_s\ge
\lambda_c$.
