Black holes surrounded by axisymmetric structures prosper in some of the most interesting sources in the universe. However, a consistent exact description of the gravitational field of these systems is still lacking. In a static case, the task reduces to Laplace equation and the fields of multiple sources follow by mere superposition. In a rotating case, non-linearity of the Einstein equations resists simple grasp, but even then the theory of completely integrable systems seems to verge on satisfactory solutions. It seeks them in terms of θ-functions on special manifolds connected – symptomatically – with the names of Riemann and Hilbert.