  We develop a general perturbative analysis on vacuum spacetimes which can be
constructed by generating manifolds of revolution around a curve, and apply it
to the Schwarzschild metric. The following different perturbations are carried
out separately: 1) Non-rotating 2-spheres are added along a plane curve
slightly deviated from the ``Schwarzschild line''; 2) General non-rotating
topological 2-spheres are added along the ``Schwarzschild line'' 3)
Slow-rotating 2-spheres are added along the ``Schwarzschild line''. For (1), we
obtain the first order vacuum solution and show that no higher order solution
exists. This linearised vacuum solution turns out however to be just a gauge
transformation of the Schwarzschild metric. For (2), we solve the general
linearised vacuum equations under several special cases. In particular, there
exist linearised vacuum solutions with signature-changing metrics that contain
closed timelike curves (though these do not correspond to adding topological
2-spheres). For (3), we find that the first order vacuum solution is equivalent
to the slowly rotating Kerr metric. This is hence a much simpler and
geometrically insightful derivation as compared to the gravitomagnetic one,
where this rotating-shells construction is a direct manifestation of the
frame-dragging phenomenon. We also show that the full Kerr however, cannot be
obtained via adding rotating ellipsoids.
