Atomic models higher up

There exists a countable structure M of Scott rank ωCK 1 where ωM 1 = ω CK 1 and where the LωCK 1 ,ω-theory of M is not ω-categorical. The Scott rank of a model is the least ordinal β where the model is prime in its Lωβ,ω-theory. Most well-known models with unbounded atoms below ωCK 1 also realize a non-principal LωCK 1 ,ω-type; such a model that preserves the Σ1-admissibility of ωCK 1 will have Scott rank ωCK 1 + 1. Makkai ([?]) produces a hyperarithmetical model of Scott rank ωCK 1 whose LωCK 1 ,ω-theory is ω-categorical. A computable variant of Makkai’s example is produced in [?]. The Makkai result taken together with the result here provide a relatively complete picture of what is possible with high Scott rank, although there are several remaining puzzles.