  Starting from the infinite set of possible master equations for the
perturbations of Schwarzschild black holes, with master functions linear in the
metric perturbations and their first-order derivatives, we show 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.
