A Theorem is proved which reduces the problem of completeness of orbits of Killing vector fields in maximal globally hyperbolic, say vacuum, space–times to some properties of the orbits near the Cauchy surface. In particular it is shown that all Killing orbits are complete in maximal developements of asymptotically flat Cauchy data, or of Cauchy data prescribed on a compact manifold.

It is demonstrated that, in the adiabatic approximation, nonequatorial circular orbits of particles in the Kerr metric ~i.e., orbits of constant Boyer-Lindquist radius! remain circular under the influence of gravitational radiation reaction. A brief discussion is given of conditions for breakdown of adiabaticity and of whether slightly noncircular orbits are stable against the growth of eccentr...

In this paper we prove that at every energy level the anisotropic problem with small anisotropy has two periodic orbits which bifurcate from elliptic orbits of the Kepler problem with high eccentricity. Moreover we provide approximate analytic expressions for these periodic orbits. The tool for proving this result is the averaging theory.