Two types of optical schemes are considered for telescopes with a fast spherical primary: the first for an afocal system, and the second for a focusing system. Both are strictly corrected for spherical aberration. In the extreme case the relative aperture of the spherical primary is 1/0.35 and the length of the system is 2.8 times smaller than the diameter of the primary. Exact analytical expressions describing a profile of the aspheric secondary are presented. The field of view of the system is highly limited by coma. It is shown that this drawback might be slightly compensated by field scanning by means of rotation of the secondary about either its vertex or the center of curvature of the fixed primary. Such schemes can be used for many different purposes, such as studies of point sources, laser-beam transformation, transmission and reception of infrared emission, and indication of telescope pointing errors.