  One of the open questions of the strong interaction theory is the existence
of the Odderon, which is an unambiguous prediction of Quantum Chromodynamics,
but still not confirmed experimentally. An alternative to probe the Odderon is
the exclusive $\eta_c$ photoproduction in hadronic collisions. As the Pomeron
exchange cannot contribute to this process, its observation would indicate the
existence of the Odderon. In this paper we estimate the $\eta_c$ production in
photon - induced interactions in hadronic collisions at the AFTER@LHC
experiment. We demonstrate that the experimental analysis of this process is
feasible in the AFTER@LHC experiment and that the observation of the $\eta_c$
production in nuclear collisions is a unambiguous signature of the Odderon.
