Given a first-order theory $T$ its model companion $T^*$ can be considered. Under relatively weak assumptions on $T$ we have that $T^*$ is the theory of the class of the existentially closed models $E(T)$ of $T$. Thus determining the model companion of a theory $T$ of amounts to axiomatizing $E(T)$. Looking at $E(T)$ for the theory of groups and the theory of commutative rings yields that a theory need not have a model companion. When a theory has a model companion the question whether it is finitely axiomatizable is of interest.
I would like to know whether determining $T^*$ and its properties can shed light on $T$ besides being interesting as such. As mentioned above determining $T^*$ amounts to determining $E(T)$, so why not doing so without considering the notion of model companion.