  We present a multiparameter generalization of the St\"ackel transform (the
latter is also known as the coupling-constant metamorphosis) and show that
under certain conditions this generalized St\"ackel transform preserves the
Liouville integrability, noncommutative integrability and superintegrability.
The corresponding transformation for the equations of motion proves to be
nothing but a reciprocal transformation of a special form, and we investigate
the properties of this reciprocal transformation.
  Finally, we show that the Hamiltonians of the systems possessing separation
curves of apparently very different form can be related through a suitably
chosen generalized St\"ackel transform.
