Existentially closed and maximal models in positive logic

It is proved that a subclass of positively existentially closed models of any finitely axiomatizable $h$-universal class in a predicate signature is axiomatizable. We construct examples suggesting the necessity of these conditions for the given subclass to be axiomatizable. The concept of an $h$-maximal model is introduced. It is shown that a subclass of $h$-maximal models of any finitely axiomatizable $h$-universal class is also finitely axiomatizable. Moreover, the set of positively existentially closed models in an $h$-universally axiomatizable class coincides with the set of positively existentially closed models in its subclass of $h$-maximal models.