  During the last few decades, there has been a growing interest in exact
solutions of Einstein equations describing razor-thin disks. Despite the
progress in the area, the analytical study of geodesic motion crossing the disk
plane in these systems is not yet so developed. In the present work, we propose
a definite vertical stability criterion for circular equatorial timelike
geodesics in static, axially symmetric thin disks, possibly surrounded by other
structures preserving axial symmetry. It turns out that the strong energy
condition for the disk stress-energy content is sufficient for vertical
stability of these orbits. Moreover, adiabatic invariance of the vertical
action variable gives us an approximate third integral of motion for oblique
orbits which deviate slightly from the equatorial plane. Such new approximate
third integral certainly points to a better understanding of the analytical
properties of these orbits. The results presented here, derived for static
spacetimes, may be a starting point to study the motion around rotating,
stationary razor-thin disks. Our results also allow us to conjecture that the
strong energy condition should be sufficient to assure transversal stability of
periodic orbits for any singular timelike hypersurface, provided it is
invariant under the geodesic flow.
