In this work we continue our study initiated in cite{GFGP} on the uniqueness properties of real solutions to the IVP associated to the Benjamin-Ono (BO) equation. In particular, we shall show that the uniqueness results established in cite{GFGP} do n
ot extend to any pair of non-vanishing solutions of the BO equation. Also, we shall prove that the uniqueness result established in cite{GFGP} under a hypothesis involving information of the solution at three different times can not be relaxed to two different times.
