Minimal elementary end extensions

Suppose that $\mathcal{M}$ is a model of PA and $\mathcal{N}$ is a countably generated elementary end extension of $\mathcal{M}$. Let ${\mathfrak X}$bethesetofsubsetsofMthatarecodedby${\mathcal N}$\mathrm{.}Then${\mathcal M}$ has a minimal elementary end extension that codes exactly the same subsets of M that $\mathcal{N}$ does iff every set that is ${\mathrm{\Pi }}_1^0$-definable in $(\mathcal{M},\mathfrak{X})$ is the union of countably many sets that are ${\mathrm{\Sigma }}_1^0$-definable.