In this work (**arXiv:1503.09187**) the constant curvature analogue on the two-dimensional sphere and the hyperbolic space of an integrable Hénon-Heiles Hamiltonian of KdV type, is revisited. The resulting integrable curved Hamiltonian depends on a parameter κ which is just the curvature of the underlying space and allows one to recover the initial Hamiltonian under the smooth flat/Euclidean limit κ→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 series of potentials associated to the curved Hamiltonian is fully constructed, and corresponds to the curved integrable analogues of homogeneous polynomial perturbations of H that are separable in parabolic coordinates. Integrable perturbations are also presented, and they can be regarded as the curved counterpart of integrable rational perturbations of the Euclidean Hénon-Heiles Hamiltonian. It is explicitly shown that the latter perturbations can be understood as the “negative index” counterpart of the curved Ramani series of potentials. Furthermore, it is shown that the integrability of the curved Hénon-Heiles Hamiltonian is preserved under the simultaneous addition of curved analogues of “positive” and “negative” families of Ramani potentials.

# A curved Hénon-Heiles system and its integrable perturbations

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