The linear and the angular momentum stored in a distribution of charges in a magnetic field

We show that it is possible to define, e.g., the z-component of the linear momentum of the system formed by a charged particle and a magnetic field if and only if the magnetic field is invariant under translations along the z-axis. Similarly, it is possible to define the z-component of the angular momentum of the system formed by a charged particle and a magnetic field if and only if the magnetic field is invariant under rotations about the z-axis.