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?


EDIT: Solovay's result is fully discussed [here](https://doi.org/10.1016/S0049-237X(08)71932-3), even though it would be nice to find a more modern account. The second question still stands.



Forcing a unique $\Delta_3^1$ generic real
Since the first question has already been answered in the EDIT at the end of the question, I will focus on the second question.


The short answer to second question is: Yes, a good source is Sy Friedman's paper *The $\Pi^1\_2$-conjecture*, Journal of the American Mathematical Society, Vol. 3, No. 4 (Oct., 1990), pp. 771-791.


In the above paper, Friedman *negatively* settled a conjecture of Robert Solovay, a conjecture that stated: Assuming the existence of $0^\sharp$ , there are no $\Pi^1\_2$ singletons $s$ such that $s \in \mathrm{L}[0^\sharp]$ and yet $0^\sharp \notin \mathrm{L} [s]$. Friedman's construction is labyrinthine; a key ingredient of it is Jensen's technique of [Coding the Universe](https://www.cambridge.org/core/books/coding-the-universe/39F95AD061AE2D5A3DFB5F7B175DA7A2), and the important related work of René David in his paper *A very absolute $\Pi^1\_2$ real singleton*, Ann. Math. Logic 23 (1982), no. 2-3, 101–120 (1983).


