Noncanonical time transformations relating finite-dimensional integrable systems

We consider dual Stäckel schemes related to each other by a noncanonical transformation of the time variable. We prove that this duality of different integrable systems arises from the multivaluedness of the Abel mapping. We construct the Lax matrices and the $r$-matrix algebras for some integrable systems on a plane. The integrable deformations of the Kepler problem and the Holt-type systems are considered in detail.