  The regular or chaotic character of orbits of stars moving in the meridional
plane (R,z) of an axially symmetric elliptical galaxy with a dense, massive
spherical nucleus and a dark matter halo component is under investigation. In
particular, we explore how the flattening of an elliptical galaxy influences
the overall orbital structure of the system, by computing in each case the
percentage of chaotic orbits, as well as the percentages of orbits composing
the main regular families. In an attempt to discriminate safely and with
certainty between regular and chaotic motion, we use the Smaller ALingment
Index (SALI) method to extensive samples of orbits obtained by integrating
numerically the basic equations of motion as well as the variational equations.
In addition, a technique which is based mainly on the field of spectral
dynamics that utilizes the Fourier transform of the time series of each
coordinate is used for classifying the regular orbits into different families
and also to recognize the secondary resonances that usually bifurcate from
them. Three cases are considered in our work: (i) the case where the elliptical
galaxy is prolate (ii) the case where a spherically symmetric elliptical galaxy
is present and (iii) the case where the elliptical galaxy has an oblate shape.
Comparison between the current results and early related work is also made.
