Utilizing an image orientation change merit function, we prove the finding of Schweitzer et al. [Appl. Opt. 37, 5190 (1998)] that only two types of optically stable reflector system exist, namely, preserving or retroreflecting. We also present an analytical method for designing optically stable reflector systems comprising multiple reflectors. It is shown that an infinite number of solutions can be obtained for systems comprising more than three reflectors. Furthermore, it is shown that by adding two parallel refracting flat boundary surfaces at the entrance and exit positions of the light ray in an optical system with multiple reflectors, an optically stable prism can be obtained.