The paper under review focuses on the investigation of two classes of generalized quadrangles: the incidence structure \({\mathcal H}(3,q^2)\) of all points and lines of a non-singular Hermitian surface in PG\((3,q^2) \) and the incidence structure \({\mathcal Q}^-(5,q)\) of all points and lines of an elliptic quadric in PG\((5,q)\). These two classes of generalized quadrangles are dual to each other. B. Segre [Ann. Mat. Pura Appl., IV. Ser. 70, 1–202 (1965; Zbl 0146.16703)] introduced regular systems on \({\mathcal H}(3,q^2)\): a regular system of order \(m\) (where \(0 < m < q+1\)) on \({\mathcal H}(3,q^2) \) is a set \(\mathcal R\) of lines of \({\mathcal H}(3,q^2)\) such that every point lies exactly on \(m\) lines of \(\mathcal R\). A regular system of order \((q+1)/2\) is called a hemisystem. For odd \(q\) every regular system on \({\mathcal H}(3,q^2)\) is a hemisystem [see loc. cit. and J. A. Thas, Geom. Dedicata 10, 135–143 (1981; Zbl 0458.51010)].
A. Bruen und J. W. P. Hirschfeld proved in 1978 [Geom. Dedicata 7, 333–353 (1978; Zbl 0394.51006)] the nonexistence of regular systems on \({\mathcal H}(3,q^2)\) for \(q\) even. The author constructs a hemisystem on \({\mathcal H}(3,q^2)\) for \(q\) odd admitting the group P\(\Omega^-_{4}(q)\) of automorphisms as well as a sporadic hemisystem on \({\mathcal H}(3,25)\) admitting the alternating group \(A_{7}\). The last section contains results about special sets of \({\mathcal H}(3,q^2)\) as introduced by Shult (not yet published): such a set \(S\) is subset of \(q^2+1\) points of \({\mathcal H}(3,q^2)\) such that each point of \({\mathcal H}(3,q^2) \setminus S\) is conjugate to zero or two points of \(S\).