  We discuss the properties of orbits within the influence sphere of a
supermassive black hole (BH), in the case that the surrounding star cluster is
nonaxisymmetric. There are four major orbit families; one of these, the pyramid
orbits, have the interesting property that they can approach arbitrarily
closely to the BH. We derive the orbit-averaged equations of motion and show
that in the limit of weak triaxiality, the pyramid orbits are integrable: the
motion consists of a two-dimensional libration of the major axis of the orbit
about the short axis of the triaxial figure, with eccentricity varying as a
function of the two orientation angles, and reaching unity at the corners.
Because pyramid orbits occupy the lowest angular momentum regions of phase
space, they compete with collisional loss cone repopulation and with resonant
relaxation in supplying matter to BHs. General relativistic advance of the
periapse dominates the precession for sufficiently eccentric orbits, and we
show that relativity imposes an upper limit to the eccentricity: roughly the
value at which the relativistic precession time is equal to the time for
torques to change the angular momentum. We argue that this upper limit to the
eccentricity should apply also to evolution driven by resonant relaxation, with
potentially important consequences for the rate of extreme-mass-ratio inspirals
in low-luminosity galaxies. In giant galaxies, we show that capture of stars on
pyramid orbits can dominate the feeding of BHs, at least until such a time as
the pyramid orbits are depleted; however this time can be of order a Hubble
time.
