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We present a multiparameter generalization of the St\"ackel transform (thelatter is also known as the coupling-constant metamorphosis) and show thatunder certain conditions this generalized St\"ackel transform preserves theLiouville integrability, noncommutative integrability and superintegrability.The corresponding transformation for the equations of motion proves to benothing but a reciprocal transformation of a special form, and we investigatethe properties of this reciprocal transformation.  Finally, we show that the Hamiltonians of the systems possessing separationcurves of apparently very different form can be related through a suitablychosen generalized St\"ackel transform.

Generalized St\"ackel Transform and Reciprocal Transformations for Finite-Dimensional Integrable Systems