The transfer of homonuclear spin polarization in a small ring system of protons which are coupled with dipole-dipole interaction has been explored experimentally. The approach was to follow the magnetization transfer from one initially polarized proton to other protons in a benzene molecule dissolved in a liquid crystal matrix. The results show that, at times up to about 100 ms, the magnetization of the initially polarized spin oscillates around a value higher than 1/6 of its initial magnetization, and the system does not approach a state close to thermodynamic equilibrium on this time scale.