  In this article we take a close look at three types of transformations usable
in the Schwarzschild black hole perturbation theory: a standard (DT), a binary
(BDT) and a generalized (GDT) Darboux transformations. In particular, we
discuss the absolutely crucial property of {\em isospectrality} of the
aforementioned transformations which guarantees that the quasinormal mode (QNM)
spectra of potentials, related via the transformation, completely coincide. We
demonstrate that, while the first two types of the Darboux transformations (DT
and BDT) are indeed isospectral, the situation is wildly different for the GDT:
it violates the isospectrality requirement and is therefore only valid for the
solutions with just one fixed frequency. Furthermore, it is shown that although
in this case the GDT does provide a relationship between two arbitrary
potentials (a short-ranged and a long-ranged potentials relationship being just
a trivial example), this relationship ends up being completely formal. Finally,
we consider frequency-dependent potentials. A new generalization of the Darboux
transformation is constructed for them and it is proven (on a concrete example)
that such transformations are also not isospectral. In short, we demonstrate
how a little, almost incorporeal flaw may become a major problem for an
otherwise perfectly admirable goal of mathematical generalization.
