There are two families of gradient-index functions of cylindrical type for which the differential equations of the rays can be completely integrated without any approximations. One of these has focusing properties within the medium and so can be considered for use in GRIN rods. Both examples can prove useful in the design of conventional types of optical systems involving gradient-index elements.
E. W. Marchand, "Ray Tracing in Cylindrical Gradient-index Media," Appl. Opt. 11, 1104-1106 (1972)