= Theory of infinitely many singleton and doubleton equivalence classes
In the language containing one binary <equivalence relation> $E$, require all classes to have size at most two and infinitely many classes of each of the two sizes. This is a <complete theory>: every countable model has countably many singleton classes and countably many doubleton classes, giving a class-by-class isomorphism; <Downward Lowenheim-Skolem theorem> then rules out a sentence separating arbitrary models. It is also an <aleph-zero-categorical theory>.
It is not finitely axiomatizable. Any finite fragment of the explicit infinitude schemes has a finite model with sufficiently many classes of each size; <compactness theorem> would make such a finite fragment suffice if any finite axiomatization existed. It is not <model-complete>: adjoining a new singleton preserves the theory, and subsequently giving that new point a partner also preserves the theory, while changing the existential partner formula at that point. Naming the singleton-class predicate removes this particular obstruction to elementary embeddings.
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