  We consider a cavity optomechanical system consisting of a Bose-Einstein
condensate (BEC) interacting with two counterpropagating traveling-wave modes
in an optical ring cavity. In contrast to the more familiar case where the
condensate is driven by the standing-wave field of a high-$Q$ Fabry-P{\'e}rot
cavity we find that both symmetric and antisymmetric collective density side
modes of the BEC are mechanically excited by the light field. In the
semiclassical, mean-field limit where the light field and the zero-momentum
mode of the condensate are treated classically the system is found to exhibit a
rich multistable behavior, including the appearance of isolated branches of
solutions (isolas). We also present examples of the dynamics of the system as
input parameters such as the frequency of the driving lasers are varied.
