Categoricity of computable infinitary theories

Computable structures of Scott rank $${\omega_1^{CK}}$$ are an important boundary case for structural complexity. While every countable structure is determined, up to isomorphism, by a sentence of $${\mathcal{L}_{\omega_1 \omega}}$$, this sentence may not be computable. We give examples, in several familiar classes of structures, of computable structures with Scott rank $${\omega_1^{CK}}$$ whose computable infinitary theories are each $${\aleph_0}$$-categorical. General conditions are given, covering many known methods for constructing computable structures with Scott rank $${\omega_1^{CK}}$$, which guarantee that the resulting structure is a model of an $${\aleph_0}$$-categorical computable infinitary theory.