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 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?

EDIT: Solovay's result is fully discussed here, even though it would be nice to find a more modern account. The second question still stands.