Focusing light rays back to the vertex of a reflecting parabolic collector: the equivalent of Dionysius ear effect in optical systems

A vertical light ray coming from infinity is reflected by a primary parabolic mirror M1 having focus in F1. At a small distance from F1 a secondary mirror M2, symmetric with respect to the vertical axis, is placed. One would like to find the analytic equation of the mirror M2, so that all rays from infinity, incident on M1, could be send back to the vertex V1 of the primary mirror M1. We denote the mirror M2 with the name of zozzaroid, after P. Zozzaro, who has first proposed the problem. Recalling the famous legend about Dionysius ear, we find that M2 has either an elliptic or an hyperbolic profile.