  We consider the St\"ackel transform, also known as the coupling-constant
metamorphosis, which under certain conditions turns a Hamiltonian dynamical
system into another such system and preserves the Liouville integrability. We
show that the corresponding transformation for the equations of motion is
nothing but the reciprocal transformation of a special form and we investigate
the properties of this transformation. This result is further applied for the
study of the $k$-hole deformations of the Benenti systems or more general seed
systems.
