  Let $(V, \Om)$ be a symplectic vector space and let $\phi: M \ra V$ be a
symplectic immersion. We show that $\phi(M) \subset V$ is (locally) an
extrinsic symplectic symmetric space (e.s.s.s.) in the sense of \cite{CGRS} if
and only if the second fundamental form of $\phi$ is parallel.
  Furthermore, we show that any symmetric space which admits an immersion as an
e.s.s.s. also admits a {\em full} such immersion, i.e., such that $\phi(M)$ is
not contained in a proper affine subspace of $V$, and this immersion is unique
up to affine equivalence.
  Moreover, we show that any extrinsic symplectic immersion of $M$ factors
through to the full one by a symplectic reduction of the ambient space. In
particular, this shows that the full immersion is characterized by having an
ambient space $V$ of minimal dimension.
