Minimal elementary end extensions

Suppose that $${\mathcal M}\models \mathsf{PA}$$M⊧PA and $${\mathfrak X} \subseteq {\mathcal P}(M)$$X⊆P(M). If $${\mathcal M}$$M has a finitely generated elementary end extension $${\mathcal N}\succ _\mathsf{end} {\mathcal M}$$N≻endM such that $$\{X \cap M : X \in {{\mathrm{Def}}}({\mathcal N})\} = {\mathfrak X}$${X∩M:X∈Def(N)}=X, then there is such an $${\mathcal N}$$N that is, in addition, a minimal extension of $${\mathcal M}$$M iff every subset of M that is $$\Pi _1^0$$Π10-definable in $$({\mathcal M}, {\mathfrak X})$$(M,X) is the countable union of $$\Sigma _1^0$$Σ10-definable sets.