  An Hermitian bounded symmetric domain in a complex vector space, given in its
circled realization, is endowed with two natural symplectic forms: the flat
form and the hyperbolic form. In a similar way, the ambient vector space is
also endowed with two natural symplectic forms: the Fubini-Study form and the
flat form. It has been shown in arXiv:math.DG/0603141 that there exists a
diffeomorphism from the domain to the ambient vector space which puts in
correspondence the above pair of forms. This phenomenon is called symplectic
duality for Hermitian non compact symmetric spaces.
  In this article, we first give a different and simpler proof of this fact.
Then, in order to measure the non uniqueness of this symplectic duality map, we
determine the group of bisymplectomorphisms of a bounded symmetric domain, that
is, the group of diffeomorphisms which preserve simultaneously the hyperbolic
and the flat symplectic form. This group is the direct product of the compact
Lie group of linear automorphisms with an infinite-dimensional Abelian group.
This result appears as a kind of Schwarz lemma.
