Remark on the non-integrability of the perturbed motion of the particle in a central field in constant curvature spaces

In the recent article [1] Kozlov and Harin generalize the motion of a particle in a central field to the case of constant curvature spaces. In this remark we show that the problem of the non-integrability of the perturbed motion in a central field on the sphere and on the Lobachevski's space is reduced to the flat case considered by Holmes and Marsden using Melnikov integrals.