  We present a new sufficient condition under which a maximal monotone operator
$T:X\tos X^*$ admits a unique maximal monotone extension to the bidual
$\widetilde T:X^{**} \rightrightarrows X^*$. For non-linear operators this
condition is equivalent to uniqueness of the extension.
  The class of maximal monotone operators which satisfy this new condition
includes class of Gossez type D maximal monotone operators, previously defined
and studied by J.-P. Gossez, and all maximal monotone operators of this new
class satisfies a restricted version of Brondsted-Rockafellar condition.
  The central tool in our approach is the $\mathcal{S}$-function defined and
studied by Burachik and Svaiter in 2000 \cite{BuSvSet02}(submission date, July
2000). For a generic operator, this function is the supremum of all convex
lower semicontinuous functions which are majorized by the duality product in
the graph of the operator.
  We also prove in this work that if the graph of a maximal monotone operator
is convex, then this graph is an affine linear subspace.
