  Simpson showed that every countable model $\mathcal{M} \models \mathsf{PA}$
has an expansion $(\mathcal{M}, X) \models \mathsf{PA}^*$ that is pointwise
definable. A natural question is whether, in general, one can obtain expansions
of a non-prime model in which the definable elements coincide with those of the
underlying model. Enayat showed that this is impossible by proving that there
is $\mathcal{M} \models \mathsf{PA}$ such that for each undefinable class $X$
of $\mathcal{M}$, the expansion $(\mathcal{M}, X)$ is pointwise definable. We
call models with this property Enayat models. In this paper, we study Enayat
models and show that a model of $\mathsf{PA}$ is Enayat if it is countable, has
no proper cofinal submodels and is a conservative extension of all of its
elementary cuts. We then show that, for any countable linear order $\gamma$, if
there is a model $\mathcal{M}$ such that $\mathrm{Lt}(\mathcal{M}) \cong
\gamma$, then there is an Enayat model $\mathcal{M}$ such that
$\mathrm{Lt}(\mathcal{M}) \cong \gamma$.
