The concept of the simplest possible reflecting anastigmat is discussed, and anastigmats consisting of four spherical mirrors are introduced in this context as being the last remaining family for which the solution set has not been thoroughly explored. Burch's "plate diagram" method is introduced and used to derive a closed-form analytical solution for four-spherical-mirror anastigmatic telescope systems. This solution is then applied to mapping the solution space for four-spherical-mirror anastigmats. Two novel systems are revealed that represent the first published instances of axially symmetrical, all-spherical anastigmatic reflecting telescope designs with concave primary mirrors. Due to the completeness of the analytic description of the solution set, it also can be stated that there are no other design variants for this class of system.