= Solution
Let $S$ be the set of all <existential sentences> entailed by $T$. Certainly every model of $T$ satisfies $S$. Conversely, take $C\models S$. Part (c) gives $A\models T$ embedded in an <elementary extension> $D$ of $C$. After identifying $A$ with its image, $D$ is a superstructure of a model of $T$. The assumed closure under superstructures implies $D\models T$. Since $C\preccurlyeq D$, it follows that $C\models T$ as well. Hence
$$
\boxed{\operatorname{Mod}(T)=\operatorname{Mod}(S),\qquad S\subseteq\exists_1.}
$$
This supplies an existential axiomatization. Conversely, an <existential sentence> is preserved under embeddings because its finite tuple of witnesses and the quantifier-free facts about them persist, explaining why this is the appropriate syntactic class.
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