The upper limit of the total cross section for the pn --> pn eta reaction has been determined near the kinematical threshold in the excess energy range from 0 to 24 MeV. The measurement was performed using the COSY-11 detector setup, a deuteron cluster target, and the proton beam of COSY with a momentum of 3.35 GeV/c. The energy dependence of the upper limit of the cross section was extracted exploiting the Fermi momenta of nucleons inside the deuteron. Comparison of the determined upper limit of the ratio R_eta = sigma(pn --> pn eta) / sigma(pp --> pp eta) with the corresponding ratio for eta-meson production does not favor the dominance of the N*(1535) resonance in the production process of the eta meson and suggests nonidentical production mechanisms for eta and eta mesons.
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