We predict the Shubnikov-de Haas oscillations of the fine structure of the cyclotron resonance near its crossing with the spin resonance in the degenerate two-dimensional Coulomb gas subjected to a strong tilted magnetic field. The spin-orbit coupling splits these resonances into three lines with the splitting gaps varying with the filling factor from v(so)p(F) to v(so)p(F)/square-root 2 at odd- and even-integer filling, respectively. This indicates reentrance of the system into a partially spin-polarized state and can be treated as manifestation of a new spin-density wave mode in it.