  The existence of static and axially symmetric regions in a Friedman-Lemaitre
cosmology is investigated under the only assumption that the cosmic time and
the static time match properly on the boundary hypersurface. It turns out that
the most general form for the static region is a two-sphere with arbitrarily
changing radius which moves along the axis of symmetry in a determined way. The
geometry of the interior region is completely determined in terms of background
objects. When any of the most widely used energy-momentum contents for the
interior region is imposed, both the interior geometry and the shape of the
static region must become exactly spherically symmetric. This shows that the
Einstein-Straus model, which is the generally accepted answer for the null
influence of the cosmic expansion on the local physics, is not a robust model
and it is rather an exceptional and isolated situation. Hence, its suitability
for solving the interplay between cosmic expansion and local physics is
doubtful and more adequate models should be investigated.
