I know Jensen developed a forcing notions in $L$ that adds a unique, minimal and $\Delta_3^1$ $L$-generic real. In his paper [Definable sets of minimal degree](https://zbmath.org/?q=an:0245.02055) he says that Solovay had already shown the consistency relative to $\mathsf{ZF}$ of $``V$ is the constructible closure of a real which is the unique solution of a $\Pi_2^1$ predicate (hence the real is $\Delta_3^1$)$"$. I searched a bit, but I couldn't find an exhaustive presentation of Solovay's result.
My questions are:

 - Is there a paper/thesis where the abovementioned result due to Solovay is explained?
 - Are there other known forcing notions adding an $L$-generic real which is the unique solution of a $\Pi_2^1$ predicate?