= Solution
With no dependence on the azimuthal or axial coordinates, <Gauss's law for magnetism> in <cylindrical coordinates> gives
$$
\frac1R\frac{d(RB_R)}{dR}=0,
\qquad B_R=\frac C R.
$$
Regularity at the axis forces $C=0$, so $B_R=0$. The radial component of the magnetostatic <Ampère-Maxwell equation> is
$$
j_R=\frac c{4\pi}(\nabla\times\mathbf B)_R
=\frac c{4\pi}\left(\frac1R\partial_\phi B_z-\partial_z B_\phi\right)=0.
$$
Consequently
$$
\boxed{B_R=j_R=0.}
$$
This is also the starting point of <cylindrical magnetostatic pressure balance>.
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