Bäcklund transformations relating different Hamilton–Jacobi equations

We discuss one of the possible finite-dimensional analogues of the general Bäcklund transformation relating different partial differential equations. We show that different Hamilton–Jacobi equations can be obtained from the same Lax matrix. We consider Hénon–Heiles systems on the plane, Neumann and Chaplygin systems on the sphere, and two integrable systems with velocity-dependent potentials as examples.