We studied some important questions in general relativity and mathematical physics mainly related to the two most important solutions of the theory of relativity - gravitational waves and black holes. In particular, the work is related to astrophysical shock waves, gravitational waves, black holes, integrable systems associated with them as well as their quantum equivalents. We studied the effects of null shells on geodesic congruences and suggested a general covariant definition of the gravitational memory effect. Thus, we studied observable effects that astrophysical shock waves can have on test particles after cataclysmic astrophysical events. We studied the geodesics of massive particles in Near Horizon Extremal Myers-Perry (NHEMP) black hole geometries. This is the space-time in the vicinity of the horizon of higher dimensional rotating black holes. Thus, this work can have applications for studying accretions of black holes. The system is also important in mathematical physics as it describes integrable (in special cases superintegrable) system, where the constants of motion are fully studied. On the other hand, the quantum counterparts of this and other integrable systems are studied as well and a new technique is suggested for geometrization of these systems.
Recent Developments in Integrable Systems and Related Topics of Mathematical Physics