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

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