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The non-relativistic absolute values and shapes of the cross sections for the process of two-photon one-electron excitation-ionization of the 1s shell of highly charged positive atomic ions Ne6+ and Ne8+ are calculated using second-order quantum-mechanical perturbation theory, taking into account the many-particle effect of relaxation of the ion residue in the field of the virtual vacancies created. The many-particle effect of stabilization of the deep vacancy when changing target from the neutral atom to the highly charged positive ions is also taken into account. The calculations are fulfilled for both linearly and circularly polarized incident photons.