In this communication, we study the use of metasurfaces comprising circular metallic patches printed on a grounded dielectric substrate to design and synthesize the double-focus generalized Luneburg lens (DF-GLL) at microwave frequencies. When illuminated by incident plane waves arriving from not just one but numerous azimuth angles, the simulated design displays two clear focal spots toward all directions, thereby exhibiting rotational symmetry which facilitates beam switching and scanning. How such circular patch elements compare with square ones is also investigated in terms of the range over which refractive indices can be synthesized as well as the bandwidths of both focal points produced by DF-GLLs composed of patches bearing those two shapes. When operated as a radiating antenna instead, for which excitation source feeds are placed at the two focal spots aligned along the azimuth angle pertaining to that intended beam direction, strong directivities toward all directions are achieved as simulated radiation patterns show. A prototype of the design was also manufactured and measurement results agree well with predictions by simulations, particularly in terms of realizing two distinct focal points toward several azimuth directions.