  Let $A$ be a densely defined simple symmetric operator in $\gH$, let
$\Pi=\bt$ be a boundary triplet for $A^*$ and let $M(\cd)$ be the corresponding
Weyl function. It is known that the Weyl function $M(\cd)$ determines the
boundary triplet $\Pi$, in particular, the pair ${A,A_0}$, where $A_0:=
A^*\lceil\ker\G_0 (= A^*_0)$, uniquely up to unitary similarity. At the same
time the Weyl function corresponding to a boundary triplet for a dual pair of
operators defines it uniquely only up to weak similarity.
  In this paper we consider symmetric dual pairs ${A,A}$ generated by $A\subset
A^*$ and special boundary triplets $\wt\Pi$ for ${A,A}$. We are interested
whether the result on unitary similarity remains valid provided that the Weyl
function corresponding to $\wt\Pi$ is $\wt M(z)= K^*(B-M(z))^{-1} K,$ where $B$
is some non-self-adjoint bounded operator in $\cH$. We specify some conditions
in terms of the operators $A_0$ and $A_B= A^*\lceil \ker(\G_1-B\G_0)$, which
determine uniquely (up to unitary equivalence) the pair ${A,A_B}$ by the Weyl
function $\wt M(\cd)$. Moreover, it is shown that under some additional
assumptions the Weyl function $M_\Pi(\cdot)$ of the boundary triplet $\Pi$ for
the dual pair $\DA$ determines the triplet $\Pi$ uniquely up to unitary
similarity. We obtain also some negative results demonstrating that in general
the Weyl function $\wt M(\cd)$ does not determine the operator $A_B$ even up to
similarity.
