I'm searching for theories that have a vaughtian pair. I've been given a hint, that $T_{RG}$ has at least one. I have also found many theorems stating in which cases a theory has no vaughtian pair, but none the other way round.
1) Can you give me a (short) proof?
2) Do you know other theories that have a vaughtian pair (with(out) proof)?
Definition of a vaughtian pair: T has a Vaughtian pair if there are two models $\mathfrak{M} \prec \mathfrak{N}$ and an L(M)-formula $\phi(x)$ such that: $\mathfrak{M} \neq \mathfrak{N}$, $\phi(\mathfrak{M})$ is infinite and $\phi(\mathfrak{M}) = \phi(\mathfrak{N})$