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We present a new method to derive the shape of the potential from the velocity field of a gas ring, or a gas disk with a flat rotation curve. The method is an extension of previous work by Binney and Teuben, and it can detect deviations from axisymmetry at the level of a few percent. The velocity field of the ring or disk is expanded into harmonics, and we present analytic expressions which relate these harmonic terms to the intrinsic parameters, and the viewing angles. We show that both the velocity field and the geometry of the ring are necessary to give complete information on the shape of the potential in the plane of the ring. The velocity field alone gives incomplete information for small ellipticities. We present new neutral hydrogen data on the H I ring around the early-type galaxy IC 2006, which was discovered by Schweizer, van Gorkom, and Seltzer (1989). The new data show that the ring is filled and has a remarkably regular velocity field. Application of our method to this gas ring shows that the halo must be close to perfectly axisymmetric. We detect a nonsignificant ellipticity of the potential of 0.012 +/- 0.026. The 95% confidence limit on the ellipticity is 0.05. This implies that the potential is nearly circular in the plane of the ring. The analysis indicates that the circular velocity is nearly constant from 0.5R(e) to 6.5R(e). We confirm that the MIL ratio in the outer parts increases (Schweizer et al. 1989). The stellar component probably has a strong disk. The data demonstrate that galaxies other than spiral galaxies have massive halos. The inferred shape of the halo can be contrasted to the strongly triaxial halos found in simulations of dissipationless halo formation. As suggested by Katz and Gunn (1991), the inclusion of baryonic matter in the simulations may be necessary to resolve this issue.

NGC 5077 is an elliptical galaxy with a disk of ionized gas along its minor axis. Observations of the structure and kinematics of this gas disk allow us to investigate both the intrinsic shape and the mass distribution of the underlying galaxy. The rotation axis of the gas disk is displaced by 23-degrees +/- 5-degrees from the apparent major axis of the galaxy, so the gas cannot lie in the equatorial plane of an axisymmetric potential. If the galaxy is triaxial and the gas disk has settled into a principal plane then there are two possible configurations, both with the gas on elliptical orbits. One is an oblate triaxial system with the gas in a plane perpendicular to the longest axis, the other is a prolate triaxial galaxy with the gas in the plane perpendicular to the shortest axis. Alternatively, the gas may lie in the warped and precessing polar ring around an oblate galaxy - in this case the gas must have been acquired recently, probably less than 10(9) yr ago. The observed rotation curve of the gas rises slowly, reaching a peak at approximately 10" from the nucleus. For circular orbits in a spherical potential, this would imply a decrease of M/L toward the center, since the core radius of the light is 1" or less. For gas on elliptic orbits, however, the precise shape of the observed rotation curve will depend on the viewing angle. If the gas lies in a warped polar ring, material further out may be seen in projection close to the center so that the observed rotation velocity is much less than the true circular speed. For NGC 5077, we construct various triaxial and oblate mass models which have constant M/L and are consistent both with the observed velocity fields of gas and stars and with the central mass-to-light ratio implied by the observed stellar velocity dispersion. Thus the slowly rising rotation curve of the gas in NGC 5077 does not require the presence of dark matter in the inner regions.