= Solution
Use the <first-order language> with one constant $c$, take $\mathcal M=(\mathbb N,c=0)$ with equality only, and let $A=\varnothing$. Set
$$
P=\operatorname{tp}^{\mathcal M}(0/\varnothing).
$$
This <complete type> contains $x=c$ and is isolated by that formula. In any <first-order structure>, at most one element can satisfy $x=c$. In every $\mathcal N\equiv\mathcal M$, its interpretation of $c$ realizes $P$, by <elementary equivalence>. Hence \b[no elementarily equivalent model has infinitely many realizations of $P$]. An <algebraic type> can have a fixed finite number of realizations in every model.
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