Common point reguli of different generalized hexagons on

In this paper, we consider any two split Cayley generalized hexagons represented on the parabolic quadric Q ( 6 , q ) and determine their common point reguli. As an application of our results we investigate which 1-systems of Q ( 6 , 3 ) that are a derivation of the exceptional spread of H ( 3 ) , see [A. De Wispelaere, J. Huizinga, H. Van Maldeghem, Ovoids and spreads of the generalized hexagon H ( 3 ) , Discrete Math. 305 (1–3) (2005) 299–311], are a spread of some hexagon on this quadric.