The motion of test particles in the gravitational fields generated by the
first four members of the infinite family of generalized Kalnajs discs, is
studied. In first instance, we analyze the stability of circular orbits under
radial and vertical perturbations and describe the behavior of general
equatorial orbits and so we find that radial stability and vertical instability
dominate such disc models. Then we study bounded axially symmetric orbits by
using the Poincare surfaces of section and Lyapunov characteristic numbers and
find chaos in the case of disc-crossing orbits and completely regular motion in
other cases
