The trajectory of all nonparaxial accelerating two-dimensional (2-D) light beams known to date cannot bend higher than a semicircle. A recent paper by Alonso and Bandres suggested the use of an additional mirror to generate an accelerating beam’s path that would be curved greater than a semicircle but less than an entire circle. We, for the first time, show how to generate a 2-D light field with its power flux circulating about a ring. Also, we consider accelerating nonparaxial asymmetric 2-D Bessel beams, which are obtained from a conventional 2-D Bessel beam by shifting its center to the complex plane. We show numerically that with increasing asymmetry parameter of the Bessel beam, its curved path becomes shorter, whereas the side-lobes are suppressed with respect to the central peak.