Quadragoni di Tits e sistemi rigati

G. Tallini has studied the generalized quadrangles satisfying the two natural conditions (a) if P and Q are points that can be joined by a line there exists a point T such that neither P and T nor Q and T can be joined, and (b) there exists a point lying on three distinct lines. Such incidence structures arise in various ways in connection with polarities. On the other hand, the author of the present paper shows that if (a) or (b) are violated in a generalized quadrangle then it must belong to a list of four fairly degenerate types of examples.