  Perturbed stationary axisymmetric isolated bodies, e.g. stars, represented by
a matter-filled interior and an asymptotically flat vacuum exterior joined at a
surface where the Darmois matching conditions are satisfied, are considered.
The initial state is assumed to be static. The perturbations of the matching
conditions are derived and used as boundary conditions for the perturbed Ernst
equations in the exterior region. The perturbations are calculated to second
order. The boundary conditions are overdetermined: necessary and sufficient
conditions for their compatibility are derived. The special case of
perturbations of spherical bodies is given in detail.
