  In this article we examine the regularity of two types of weak solutions to a
Monge-Amp\`ere type equation which emerges in a problem of finding surfaces
that refract coaxial light rays emitted from source domain and striking a given
target after refraction. Historically, ellipsoids and hyperboloids of
revolution were the first surfaces to be considered in this context. The
mathematical formulation commences with deriving the energy conservation
equation for sufficiently smooth surfaces, regarded as graphs of functions to
be sought, and then studying the existence and regularity of two classes of
suitable weak solutions constructed from envelopes of hyperboloids or
ellipsoids of revolution. Our main result in this article states that under
suitable conditions on source and target domains and respective intensities
these weak solutions are locally smooth.
