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.