A note on standard systems and ultrafilters

Let $(M,\text{scott}X)\models \text{ACA}$ be such that $P_{\text{scott}}X$, the collection of all unbounded sets in $\text{scott}X$, admits a definable complete ultrafilter and let $T$ be a theory extending first order arithmetic coded in $\text{scott}X$ such that $M$ thinks $T$ is consistent. We prove that there is an end-extension $N \models T$of$M$suchthatthesubsetsof$M$codedin$N$ are precisely those in $\text{scott}X$. As a special case we get that any Scott set with a definable ultrafilter coding a consistent theory $T$ extending first order arithmetic is the standard system of a recursively saturated model of $T$.