A curved Henon-Heiles system and its integrable perturbations

The constant curvature analogue on the two-dimensional sphere and the hyperbolic space 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, is revisited. The resulting integrable curved Hamiltonian, $\mathcal{H}_\kappa$, depends on a parameter $\kappa$ which is just the curvature of the underlying space and allows one to recover $\mathcal{H}$ under the smooth flat/Euclidean limit $\kappa\to 0$. This system can be regarded as an integrable cubic perturbation of a specific curved $1:2$ anisotropic oscillator, which was already known in the literature. The Ramani-Dorizzi-Grammaticos (RDG) series of potentials associated to $\mathcal{H}_\kappa$ is fully constructed, and corresponds to the curved integrable analogues of homogeneous polynomial perturbations of $\mathcal{H}$ that are separable in parabolic coordinates. Integrable perturbations of $\mathcal{H}_\kappa$ are also fully presented, and they can be regarded as the curved counterpart of integrable rational perturbations of the Euclidean Hamiltonian $\mathcal{H}$. It will be explicitly shown that the latter perturbations can be understood as the "negative index" counterpart of the curved RDG series of potentials. Furthermore, it is shown that the integrability of the curved H\'enon-Heiles Hamiltonian $\mathcal{H}_\kappa$ is preserved under the simultaneous addition of curved analogues of "positive" and "negative" families of RDG potentials.