Since under a gauge transformation the vector potential transforms with the gradient of a scalar function, the curl of the vector potential is a gauge-independent quantity. We have shown here, for the particular case of two noninteracting electrons in an external Coulomb potential, that the vector potential has non-vanishing curl. There- fore, it cannot be gauged away in this situation. Whether this property holds valid in more realistic cases, as for instance in the presence of electron-electron interaction, will be investigated in future work.