= Solution
The theory $\operatorname{Th}(\mathbb Z,+,0)$ is not <aleph-zero-categorical theory>[aleph-zero-categorical]. Introduce a constant $c$ and add the formulas
$$
c\ne0,
\qquad
\exists y\;(ny=c)quad(n=1,2,\ldots).
$$
Every finite subset is realized in $\mathbb Z$ by taking $c$ to be a nonzero common multiple of the finitely many displayed integers. The <compactness theorem> therefore gives a model of $\operatorname{Th}(\mathbb Z,+,0)$ containing a nonzero <infinitely divisible element of an abelian group>. The <Downward Lowenheim-Skolem theorem> gives such a model that is countable.
No nonzero integer is divisible by every positive integer, so this countable model is not isomorphic to $\mathbb Z$. This is the <nonstandard model of the additive integers> obstruction to categoricity.
Solved by gpt-5.6-sol high.
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