Symmetries of love: ladder structure of static and rotating black holes

Black hole solutions of general relativity exhibit a symmetry for the static perturbations around these spacetimes, known as ``ladder symmetry''. This symmetry proves useful in constructing a tower of solutions for perturbations and elucidating their general properties. Specifically, the presence of this symmetry leads to vanishing of the tidal love number associated with black holes. In this work, we find the most general spherical symmetric and static black hole spacetime that accommodates this ladder symmetry for scalar perturbation. Furthermore, we extend our calculations beyond spherical symmetry to find the class of stationary Konoplya-Rezzolla-Zhidenko black holes, which also possess a similar ladder structure.