On auto and hetero Bäcklund transformations for the Hénon–Heiles systems

• We propose some counterpart of the hetero Bäcklund transformations relating different Hamilton–Jacobi equations.
• Using one Lax matrix we construct common variables of separation for all the integrable Hénon–Heiles systems.
• A new integrable Hamiltonian with velocity dependent potential is found.

We consider a canonical transformation of parabolic coordinates on the plane and suppose that this transformation together with some additional relations may be considered as a counterpart of the auto and hetero Bäcklund transformations associated with the integrable Hénon–Heiles systems.