  We investigate the orbital structure of a model triaxial star cluster,
centered around a supermassive black hole (BH), appropriate to galactic nuclei.
Sridhar and Touma (1999) proved that the presence of the BH enforces some
regularity in the dynamics within the radius of influence of the BH. We employ
their averaging method to reduce the degrees of freedom from three to two.
Numerical orbit integrations, together with Poincar\'e surfaces of section
allow us to draw a global portrait of the orbital structure; in our
calculations we employ a model cluster potential that is triaxial and harmonic.
The averaged dynamics of the axisymmetric case is integrable, and we present a
detailed comparison of orbits in oblate and prolate axisymmetric potentials.
Both cases support resonant orbits with fixed values of eccentricity,
inclination, and periapse, whose lines of nodes rotates steadily. We then
systematically explore significantly triaxial potentials, possessing small
oblateness, or prolateness. Resonant orbits and their families are studied both
numerically, and through secular perturbation theory. Chaos appears to be
suppressed for all the cases we studied, and we obtain effective third
integrals. Some of the orbits appear to reinforce the shape of the potential;
we provide phase space, as well as real space portraits of these orbits. A
particularly promising resonant orbit exists in highly prolate, triaxial
potentials.
