We show that if an electromagnetic field is invariant under translations or rotations, three of the six components of the field can be expressed in terms of a (gauge-invariant) scalar potential which is also invariant under these transformations. This scalar potential appears in the constant of motion associated with this symmetry for a charged test particle in this field. We also show that the Cartesian components of the electromagnetic field can be combined to form two SO(2, 1) vectors
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G. F. Torres del Castillo (2024) studied this question.
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