  The study of the matching of stationary and axisymmetric spacetimes with
Friedmann-Lemaitre-Robertson-Walker spacetimes preserving the axial symmetry is
presented. We show, in particular, that any orthogonally transitive stationary
and axisymmetric region in FLRW must be static, irrespective of the matter
content. Therefore, previous results on static regions in FLRW cosmologies
apply. As a result, the only stationary and axisymmetric vacuum region that can
be matched to a (non-static) FLRW spacetime is a spherically symmetric region
of Schwarzschild. This constitutes another uniqueness result for the
Einstein-Straus model (as well as its Oppenheimer-Snyder counterpart), and
hence another indication of its unsuitability as an answer to the influence of
the cosmic expansion on local physics.
