  The billiard systems within quadrics, playing the role of discrete analogues
of geodesics on ellipsoids, are incorporated into the theory of integrable
quad-graphs. An initial observation is that the Six-pointed star theorem, as
the operational consistency for the billiard algebra, is equivalent to an
integrabilty condition of a line congruence. A new notion of the
double-reflection nets as a subclass of dual Darboux nets associated with
pencils of quadrics is introduced, basic properies and several examples are
presented. Corresponding Yang-Baxter maps, associated with pencils of quadrics
are defined and discussed.
