Common point reguli of different generalized hexagons on Q(6,q)

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.