= Solution
Take the stated static <vector potential> with zero <scalar potential>. In the laboratory,
$$
\mathbf E=0,\qquad
\mathbf B=\nabla\times\mathbf A
=\bigl(A_0k_u\sin(k_uz),0,0\bigr).
$$
Let $S'$ move along $z$ at the <electron>'s reference <velocity> $v$. The inverse transformation is $z=\gamma(z'+vt')$. The <Lorentz transformation of electromagnetic fields> gives
$$
\boxed{B'_x=\gamma A_0k_u\sin[\gamma k_u(z'+vt')],\qquad
E'_y=\gamma vA_0k_u\sin[\gamma k_u(z'+vt')],}
$$
with all other components zero. The sign follows from $\mathbf v\times\mathbf B=vB_x\mathbf e_y$.
Although an observer fixed in $S'$ sees an oscillating field, this is not an exact source-free propagating <electromagnetic wave>. The phase covector has $\omega'=-\gamma vk_u$, $k'_z=\gamma k_u$, so its squared norm is
$$
\boxed{\left(\frac{\omega'}c\right)^2-(k'_z)^2=-k_u^2<0.}
$$
It is the boost of a static spatial grating, not a null vacuum wave. Equivalently, $E'/B'=v<c$ and $B'^2-E'^2/c^2=B^2$ is nonzero away from nodes; a vacuum <plane wave> would have vanishing <electromagnetic field invariants>. In the ultrarelativistic limit it can be a useful nearly wave-like approximation along the <electron>'s path, but the exact distinction remains.
A charged <electron> cannot actually maintain the given constant <velocity> through this nonzero transverse field without an external constraint: the <Lorentz force> deflects it. The calculation therefore uses the prescribed straight reference trajectory, a weak-deflection approximation, or an instantaneously comoving inertial frame. A globally fixed <electron> rest frame is not available for its exact deflected motion.
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