  We introduce the notion of a generalized Jung factor: a II$_1$ factor $M$ for
which any two embeddings of $M$ into its ultrapower $M^{\mathcal U}$ are
equivalent by an automorphism of $M^{\mathcal U}$. We show that $\mathcal R$ is
not the unique generalized Jung factor but is the unique $\mathcal R^{\mathcal
U}$-embeddable generalized Jung factor. We use model-theoretic techniques to
obtain these results. Integral to the techniques used is the result that if $M$
is elementarily equivalent to $\mathcal R$, then any elementary embedding of
$M$ into $\mathcal R^{\mathcal U}$ has factorial relative commutant. This
answers a long-standing question of Popa for an uncountable family of II$_1$
factors. We also provide new examples and results about the notion of super
McDuffness, which is a strengthening of the McDuff property for II$_1$ factors.
