  We show that certain non-special but weakly special threefolds $X$
constructed by Bogomolov-Tschinkel enjoy strong complex hyperbolicity
properties: their entire curves are algebraically degenerate and lie either on
a fixed divisor or on the fiber of the unique elliptic fibration on $X$. These
properties are consistent with the conjectural link between hyperbolicity and
arithmetics of projective manifolds. The arithmetic counterpart of this
hyperbolicity property (the non-potential density of $X$, object of two
conflicting conjectures) remains however open.
