This question is not about the solutions but much rather about the equations we write in GR for a spherically symmetric, static vacuum 4D spacetime.
The Einstein equations are $$G_{\mu\nu}=0\;\;\;\Rightarrow\;\;\; g^{\mu\nu}G_{\mu\nu}=0\;\;\;\Rightarrow \;\;\;R=0$$ so that in conclusion we can solve $$R_{\mu\nu}=0 \;\;\;\text{or}\;\;\;R^{\mu}_{\nu}=0$$ Moreover, we can choose the coordinates so that the metric takes the form $$ds^2=-f(r)\,dt^2+h^{-1}(r)\,dr^2+r^2\!\left(d\theta^2+\sin^2\!\theta \,d\varphi^2\right)$$ In this way when we write explicitly the equations $R^{\mu}_{\nu}=0$ we get three non trivial equations, namely: $$R^{0}_0=0\;\;,\qquad R^r_r=0\;\;,\qquad R^{\theta}_{\theta}\equiv R^{\varphi}_{\varphi}=0$$
Using the first two equations and the formal identity $R^{\theta}_{\theta}\equiv R^{\varphi}_{\varphi}$ , we can write the third simply as $R=0$.
My question is what is the meaning of this third equation, when the variables are just the two functions $f(r)\;$ and $\;h(r)$? Does it act like a constraint or is there any way to get around it? and what does it tell us about GR in spherically symmetric and static space-times?
I would guess that it corresponds to a relation we impose between $f$, $h$ (and the coordinate $r$) when we write the solid angle part as in flat space, but I don't really understand its role.