This paper discusses non-rotationally symmetric geodesic lens antennas. By defining the shape of the lens footprint, additional degrees of freedom are introduced into the design process, allowing for further tailoring of the electromagnetic performance of the lens while simultaneously reducing the volume occupied by the lens. An in-house ray-tracing model for geodesic lenses has been modified to analyze this class of lenses. Examples showing the effect of modifying the lens profile are presented, with a more detailed discussion on profile presenting improved side-lobe levels at the extreme ports. The examples presented operate at 30 GHz.