On isomorphism of integrable cases of the Euler equations on the bi-hamiltonian manifolds $e(3)$ and $so(4)$

The Poisson maps between the Clebsch model and the Schottky system, two Steklov systems, the Kowalevski top and the Neumann system are considered. We prove that these non-canonical transformations of variables are the twisted Poisson maps, which completely define the corresponding pairs of integrable systems.