Geodesic chaos around quadrupolar deformed centers of attraction

The exact solution to the Einstein equations that represent a static axially symmetric source deformed by an internal quadrupole is considered. The Poincaré section method is used to study numerically the geodesic motion of test particles, for prolate quadrupole deformations, we find chaotic motions contrary to the oblate case where only regular motion is found. We also consider the metric that represents a rotating black hole deformed by a quadrupole term. This metric is obtained as a two-soliton solution in the context of Belinsky-Zakharov inverse scattering method. The stability of geodesics depends strongly on the relative direction of the spin of the center of attraction and the angular momentum of the test particle.