= Solution
Take the parameter-free <complete type>
$$
p(x)=\operatorname{tp}^{\mathcal Z^*}(c/\varnothing)
$$
of the nonstandard prime constructed in part (i). It is <first-order consistent> and complete because it consists of exactly the <logical formulas> true of an actual element in a <first-order model> of $T$. It contains $x\ne\bar a$ for every <integer> $a$, and consequently no element of $\mathcal Z$ realizes it. Thus
$$
\boxed{p\text{ is a complete type over }T\text{ omitted in }\mathcal Z.}
$$
It is necessarily a <nonprincipal complete type>. If a <first-order consistent> <logical formula> isolated it, completeness of $T$ would force the existential closure of that <logical formula> to hold in every <first-order model> of $T$, including $\mathcal Z$; a witness there would realize the <partial type>, a <contradiction>. This connects the explicit omission with the hypothesis of the <omitting types theorem>.
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