Vaught's conjecture for quite o-minimal theories

We study Vaught's problem for quite o-minimal theories. Quite o-minimal theories form a subclass of the class of weakly o-minimal theories preserving a series of properties of o-minimal theories. We investigate quite o-minimal theories having fewer than $2^{\omega }$ countable models and prove that the Exchange Principle for algebraic closure holds in any model of such a theory and also we prove binarity of these theories. The main result of the paper is that any quite o-minimal theory has either $2^{\omega }$ countable models or $6^a{3}^b$ countable models, where a and b are natural numbers.