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The exact solution of the motion of a charged particle driven by an electromagnetic pulse shape of arbitrary polarization, propagating along the direction of a static and homogeneous external magnetic field is obtained from the relativistic force equation. The method of solution allows to solve, in terms of the invariant phase, the trajectory of the particles driven by an electromagnetic pulse of arbitrary amplitude modulated by a Gaussian-like envelope. The solutions reported in this work are of concern in the interaction physics of laser-produced plasmas, to study, for instance, the mechanisms of particle acceleration, plasma heating and radiative effects.