Hidden symmetries in the dynamics of perturbed Schwarzschild Black Holes

There are two important physical processes around black holes that can be well described using relativistic perturbation theory: Scattering of electromagnetic and gravitational waves (and other fields) and quasinormal mode oscillations that take place, for instance, after the coalescence of a black hole binary. It is well-known that these physical processes can be described in terms of  gauge-invariant master functions. We have analyzed the space of all the possible master functions for the case of non-rotating black holes and we find two branches of solutions. One branch includes the known results: In the odd-parity case, the most general master function is an arbitrary linear combination of the Regge-Wheeler and the Cunningham-Price-Moncrief master functions whereas in the even-parity case it is an arbitrary linear combination of the Zerilli master function and another master function that is new to our knowledge. The other branch is very different since it includes an infinite collection of potentials which in turn lead to an independent collection master of functions which depend on the potential.  We also find that of all them are connected via Darboux transformations. These transformations preserve physical quantities like the quasinormal mode frequencies and the infinite hierarchy of Korteweg-de Vries conserved quantities, revealing a new hidden symmetry in the description of the perturbations of Schwarzschild black holes: Darboux covariance.