  The exact solution to the Einstein equations that represents a static axially
symmetric source deformed by an internal quadrupole is considered. By using the
Poincare section method we numerically study the geodesic motion of test
particles. For the prolate quadrupolar deformations we found chaotic motions
contrary to the oblate case where only regular motion is found. We also study
the metric that represents a rotating black hole deformed by a quadrupolar
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 test particle angular momentum. The rotation does not alter
the regularity of geodesic motions in the oblate case, i.e., the orbits in this
case remain regular. We also employ the method of Lyapounov characteristic
numbers to examine the stability of orbits evolving around deformed nonrotating
centers of attraction. The typical time to observe instability of orbits is
analyzed.
