On translation spreads of H(q)

Using the connection between translation spreads of $H(q)$ and linear sets (see [I. Cardinali, G. Lunardon, O. Polverino, R. Trombetti: Translation Spreads of the Classical Generalized Hexagon, {\it European J. Combin.}, {\bf 23} (2002), 367--376] and [G. Lunardon, O. Polverino: On the twisted cubic of $PG(3,q)$, {\it J. Algebr. Comb.}, {\bf 18} (2003), 255--262. ]), some results on the existence of translation spreads of $H(q)$ are given, improving classification results contained in [Offer: {\it Spreads and Ovoids of the Split Cayley Hexagon.} Ph.D. Thesis, The University of Adelaide, Adelaide (Au) 2000] and in [G. Bonoli, O. Polverino: The twisted cubic of $PG(3,q)$ and translation spreads of $H(q)$, {\it Discrete Mathematics}, {\bf 296} (2005), 129--142].