Omitting theories of undefinable sets
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Suppose $M$ is countable recursively saturated and $X$ is an undefinable subset of $M$. Is there a countable recursively saturated $N$ such that $N$ is an elementary end extension of $M$, and if $Y \subseteq M$ is coded in $N$, then $(M,Y) \not\equiv (M,X)$?
The answer if `yes' is either $(M,X)\not\models PA^*$ or ${\rm Th}(M,X)\notin {\rm SSy}(M)$.
This problem is listed in Kossak, Roman; Schmerl, James H. The structure of models of Peano arithmetic. Oxford Logic Guides, 50. Oxford Science Publications. The Clarendon Press, Oxford University Press, Oxford, 2006. xiv+311, but, unfortunately, with many typos.
