We theoretically investigate the properties of an electron energy spectrum in a double GaAs-Al0.3Ga0.7As quantum ring by using the effective mass and adiabatic approximations, together with a realistic description of the confining potential profile, which is assumed to be deformed due to the application of an intense nonresonant laser field. The effects of the applied magnetic field and spin-orbit interaction are included. We discuss the features of the lowest confined energy levels under a variation of magnetic field strengths and intense laser parameters. The influence of this external probe on the linear optical absorption response associated with interlevel transitions is analyzed by considering both the presence and absence of spin-orbit effects.