A solar concentrator with a highly asymmetric acceptance cone is investigated. Concentrating photovoltaic systems require dual-axis sun tracking to maintain nominal concentration throughout the day. In addition to collecting direct rays from the solar disk, which subtends ∼0.53 degrees, concentrating optics must allow for in-field tracking errors due to mechanical misalignment of the module, wind loading, and control loop biases. The angular range over which the concentrator maintains <90% of on-axis throughput is defined as the optical acceptance angle. Concentrators with substantial rotational symmetry likewise exhibit rotationally symmetric acceptance angles. In the field, this is sometimes a poor match with azimuth-elevation trackers, which have inherently asymmetric tracking performance. Pedestal-mounted trackers with low torsional stiffness about the vertical axis have better elevation tracking than azimuthal tracking. Conversely, trackers which rotate on large-footprint circular tracks are often limited by elevation tracking performance. We show that a line-focus concentrator, composed of a parabolic trough primary reflector and freeform refractive secondary, can be tailored to have a highly asymmetric acceptance angle. The design is suitable for a tracker with excellent tracking accuracy in the elevation direction, and poor accuracy in the azimuthal direction. In the 1000X design given, when trough optical errors (2mrad rms slope deviation) are accounted for, the azimuthal acceptance angle is +/-1.65°, while the elevation acceptance angle is only +/-0.29°. This acceptance angle does not include the angular width of the sun, which consumes nearly all of the elevation tolerance at this concentration level. By decreasing the average concentration, the elevation acceptance angle can be increased. This is well-suited for a pedestal alt-azimuth tracker with a low cost slew bearing (without anti-backlash features).