An integrable Henon-Heiles system on the sphere and the hyperbolic plane

We construct a constant curvature analogue on the two-dimensional sphere ${\mathbf S}^2$ and the hyperbolic space ${\mathbf H}^2$ of the integrable H\'enon-Heiles Hamiltonian $\mathcal{H}$ given by $$ \mathcal{H}=\dfrac{1}{2}(p_{1}^{2}+p_{2}^{2})+ \Omega \left( q_{1}^{2}+ 4 q_{2}^{2}\right) +\alpha \left( q_{1}^{2}q_{2}+2 q_{2}^{3}\right) , $$ where $\Omega$ and $\alpha$ are real constants. The curved integrable Hamiltonian $\mathcal{H}_\kappa$ so obtained depends on a parameter $\kappa$ which is just the curvature of the underlying space, and is such that the Euclidean H\'enon-Heiles system $\mathcal{H}$ is smoothly obtained in the zero-curvature limit $\kappa\to 0$. On the other hand, the Hamiltonian $\mathcal{H}_\kappa$ that we propose can be regarded as an integrable perturbation of a known curved integrable $1:2$ anisotropic oscillator. We stress that in order to obtain the curved H\'enon-Heiles Hamiltonian $\mathcal{H}_\kappa$, the preservation of the full integrability structure of the flat Hamiltonian $\mathcal{H}$ under the deformation generated by the curvature will be imposed. In particular, the existence of a curved analogue of the full Ramani-Dorizzi-Grammaticos (RDG) series $\cal{V}_{n}$ of integrable polynomial potentials, in which the flat H\'enon-Heiles potential can be embedded, will be essential in our construction. Such infinite family of curved RDG potentials $\cal{V}_{\kappa, n} $ on ${\mathbf S}^2$ and ${\mathbf H}^2$ will be also explicitly presented.