Integrable lattice geometry and pencils of quadrics

The talk consists of two parts which both deal with pencils of quadrics and integrable quad graphs. In the first part we connect the class of discriminantly separable polynomials in three variables of degree two in each, which we introduced recently, with integrable quad-graphs in the sense of Adler, Bobenko and Suris, and with pencils of conics. We present a classification of such polynomials and compare it with the ABS classification of integrable quad graphs. In the second part, we start with the billiard algebra, associated with billiard systems within pencils of quadrics and our recent "six-pointed star theorem", which is an operational consistency for the billiard algebra operation. It can be interpreted as a consistency condition for a line congruence. Developing this subject, we propose a new notion of "double reflection" nets as a subclass of dual Darboux nets. The results from the first part are joint with Katarina Kukic, and from the second part with Milena Radnovic.