A theoretical study of the relativistic cyclotron motion occurring in a uniform magnetic field and an oscillating electric field of arbitrary polarization is performed, which aims at determining the effect of the ellipticity and the strength of the electric field upon the integrability or nonintegrability of the system. Unless a circularly polarized electric field is used, the cyclotron system is nonintegrable and displays stochastic behavior in the region where resonance islands overlap. It is found, however, that the stochastic layers become increasingly thin as the polarization angle is moved closer coward pi/2 (circular polarization). If the polarization angle is held fixed and the electric field amplitude is increased, the Kolmogorov-Arnold-Moser curve a separating the resonance islands experience a reconnection process through which the islands are topologically rearranged. When the rearrangement is accomplished, the phase space is occupied mostly by regular trajectories. [S1063-651X(99)02710-5].