Резюме: 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 triaxial. There are four major orbit families; one of these, the pyramid orbits, have the interesting property that they can approach arbitrarily close to the BH. In the limit of weak triaxiality, we show that the pyramid orbits are integrable; the motion can be described as a two-dimensional oscillation 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 as a mechanism for supplying matter to BHs. We derive expressions for the capture rate, including the effects of general relativistic precession which implies an upper limit to the eccentricity. We show that the capture rate from pyramid orbits can greatly exceed the collisional capture rate, particularly in giant galaxies, at least until such a time as the pyramid orbits are depleted; however this time can be of order a Hubble time. Pyramid orbits may dominate the production rate of extreme-mass-ratio inspirals in giant galaxies. Our results provide new insight into the mechanism of resonant relaxation, particularly as it applies to eccentric orbits.