In past decades black holes have become standard objects used both in theoretical research and as astrophysical sources. They have been observed by different methods, investigated from many points of view and generalized to various situations. In this project we will be interested in rotating black holes with spherical topology of horizon, with possible additional sources on the axis, generalized to an arbitrary dimension. These are described by so called Kerr-NUT-(A)dS spacetimes. The most of their properties stems from their high symmetry . It has been demonstrated that this symmetry is encoded by the principal tensor (the non-degenerated closed conformal Killing-Yano tensor of rank 2). It guarantees the existence of the tower of Killing tensors and Killing vectors, which imply that the geodesic motion is integrable and the basic field equations are separable.
The separability has been shown for the scalar, Dirac, electromagnetic and massive vector field. It is still an open problem for the gravitational perturbations. Also an interpretation of all modes for the electromagnetism is still not completely finished . One of the tasks in this research would be clarification of the separability for the electromagnetic field: e.g., a comparison with the standard methods in four dimension, limits to known cases of the Schwarzschild and Minkowski spacetime, etc.
Another open problem of higher dimensional black hole is the absence of a charged solution, as well as of an analogy of the C-metric solution (accelerated black holes). In four dimensions it is known that these solutions are of the Petrov type D, but the C-metric does not possesses the principal tensor, just the conformal Killing-Yano tensor. Therefore, it would be interesting to look for higher-dimensional metrics having just the conformal Killing-Yano tensor and try to employ such metrics for identifications of a solution with accelerated black holes.