  Let $\mathbb{M}$ be the monster model of a complete first-order theory $T$.
If $\mathbb{D}$ is a subset of $\mathbb{M}$, following D. Zambella we consider
$e(\mathbb{D})=\{\mathbb{D}^\prime\mid (\mathbb{M},\mathbb{D})\equiv
(\mathbb{M},\mathbb{D}^\prime)\}$ and $o(\mathbb{D})=\{\mathbb{D}^\prime\mid
(\mathbb{M},\mathbb{D})\cong (\mathbb{M},\mathbb{D}^\prime)\}$. The general
question we ask is when $e(\mathbb{D})=o(\mathbb{D})$ ? The case where
$\mathbb{D}$ is $A$-invariant for some small set $A$ is rather straightforward:
it just mean that $\mathbb{D}$ is definable. We investigate the case where
$\mathbb{D}$ is not invariant over any small subset. If T is geometric and
$(\mathbb{M},\mathbb{D})$ is an $H$-structure (in the sense of A. Berenstein
and E. Vassiliev) or a lovely pair, we get some answers. In the case of
$SU$-rank one, $e(\mathbb{D})$ is always different from $o(\mathbb{D})$. In the
o-minimal case, everything can happen, depending on the complexity of the
definable closure.
