  The $\Delta S = 2$ $b \to \bar d ss$ transition proceeds via the box -
diagram in the Standard Model with a branching ratio calculated to be below
$10^{-11}$, thus providing an appropriate testing ground for physics beyond the
Standard Model. We analyze the $\Delta S = 2$ two - body $B^- \to K^{*-} \bar
K^{*0}$, $B^- \to K^- \bar K^{0}$, $B^- \to K^{*-} \bar K^{0}$ and $B^- \to
K^{-} \bar K^{*0}$ exclusive decays which are driven by the $b \to \bar d ss$
transition, both in the Standard Model and in several extensions of it. The
models considered are the minimal supersymmetric model with and without ${\cal
R}$ parity conservation and two Higgs doublet models. All four modes are found
to have a branching ratio of the order of $10^{-13}$ in the Standard Model,
while the expected branching ratio in the different extensions vary between
$10^{-9} - 10^{-6}$.
