In this paper (arXiv:1411.2033) we construct a constant curvature analogue on the two-dimensional sphere and the hyperbolic space of the integrable Hénon-Heiles Hamiltonian of KdV type. The curved integrable Hamiltonian so obtained depends on a real parameter which is just the curvature of the underlying space, and is such that the Euclidean Hénon-Heiles system is smoothly obtained in the zero-curvature limit. On the other hand, the Hamiltonian 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 Hénon-Heiles Hamiltonian, the preservation of the full integrability structure of the flat Hamiltonian under the deformation generated by the curvature will be imposed. In particular, the existence of a curved analogue of the full Ramani series of integrable polynomial potentials, in which the flat Hénon-Heiles potential can be embedded, will be essential in our construction. Such infinite family of curved Ramani potentials will be also explicitly presented.