  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 not 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.
