  The problem of constructing global models describing isolated axially
symmetric rotating bodies in equilibrium is analyzed. It is claimed that,
whenever the global spacetime is constructed by giving boundary data on the
limiting surface of the body and integrating Einstein's equations both inside
and ouside the body, the problem becomes overdetermined. Similarly, when the
spacetime describing the interior of the body is explicitly given,the problem
of finding the exterior vacuum solution becomes overdetermined. We discuss in
detail the procedure to be followed in order to construct the exterior vacuum
field created by a given but arbitrary distribution of matter. Finally, the
uniqueness of the exterior vacuum gravitational field is proven by exploiting
the harmonic map formulation of the vacuum equations and the boundary
conditions prescribed from the matching.
