Simultaneous Separation for the Neumann and Chaplygin Systems

The Neumann and Chaplygin systems on the sphere are simultaneously separable in variables obtained from the standard elliptic coordinates by the proper Bäcklund transformation. We also prove that after similar Bäcklund transformations other curvilinear coordinates on the sphere and on the plane become variables of separation for the system with quartic potential, for the Hénon–Heiles system and for the Kowalevski top. This allows us to speak about some analog of the hetero Bäcklund transformations relating different Hamilton–Jacobi equations.