  We explore the orbital dynamics of a realistic three dimensional model
describing the properties of a disk galaxy with a spherically symmetric central
dense nucleus and a triaxial dark matter halo component. Regions of phase space
with regular and chaotic motion are identified depending on the parameter
values for triaxiality of the dark matter halo and for breaking the rotational
symmetry. The four dimensional Poincar\'e map of the three degrees of freedom
system is analyzed by a study of its restriction to various two dimensional
invariant subsets of its domain.
