Towards gravitating discs around stationary black holes

This article outlines the search for an exact general relativistic description of the exterior (vacuum) gravitational field of a rotating spheroidal black hole surrounded by a realistic axially symmetric disc of matter. The problem of multi-body stationary spacetimes is first exposed from the perspective of the relativity theory and astrophysics, listing the basic methods employed and results obtained. Then, basic formulas for stationary axisymmetric solutions are summarized. Remaining sections review what we have learnt with Miroslav Zacek and Tomas Zellerin about certain static and stationary situations recently. Although the survey part is quite general, the list of references cannot be complete. Our main desideratum was the informative value rather than originality – novelties have been preferred, mainly reviews and those with detailed introductions.