The article mentions "few known exact solutions". There are actually quite a number. Most are from after 1950.<p>Cambridge University Press has two good books:
Exact Spacetimes in Einstein's General Relativity (Griffith & Podolsky)
- good readable (if you speak GR) account of the more common solutions
Exact Solutions of Einstein's Feild Equations (Stephani, Kramer, MacCallum, Hoenstelaers, Herlt)
- encyclopedic, authoritative and mathematical. In my case an aspirational purchase
For anyone interested in this topic: there are commentaries on this paper as well as Gödel's own lecture on this in Volume III of his Collected Works.