  In the present paper we extend the multiparameter coupling constant
metamorphosis, also known as the generalized St\"ackel transform, from
Hamiltonian dynamical systems to general finite-dimensional dynamical systems
and ODEs. This transform interchanges the values of integrals of motion with
the parameters these integrals depend on but leaves the phase space coordinates
intact. Sufficient conditions under which the transformation in question
preserves integrability and a simple formula relating the solutions of the
original system to those of the transformed one are given.
