In this thesis we consider the motion of a charged particle in strong electromagnetic background fields, having in mind applications to state-of-the-art high-power lasers. To find solutions of the Lorentz force equation of motion, we make use of Noether's theorem to identify conserved quantities of the charge dynamics. We will explain how, given enough symmetries, a dynamical system becomes integrable or, with a maximum number of conserved quantities, maximally superintegrable. Beginning with charged particles in vector background fields, we shall show that the relevant symmetry group is the Poincaré group. The dynamics for constant and univariate fields is classified, and their integrability properties are clarified. We then move on to the problem of a particle in a scalar background field and show that the symmetry group is extended to the conformal group. We then present examples of fields which include Poincaré, dilation and special conformal symmetries leading to varying extents of integrability.