Countably categorical and autostable Boolean algebras with distinguished ideals

We study countable Boolean algebras with finitely many distinguished ideals (countable $I$-algebras) whose elementary theory is countably categorical, and autostable $I$-algebras which form their subclass. We propose a new characterization for the former class that allows to answer a series of questions about the structure of countably categorical and autostable $I$-algebras.