= Solution
Introduce the squared poloidal <Alfvén number>
$$
M_A^2=\frac{\mu_0\rho u_p^2}{B_p^2}
=\frac{\mu_0k^2}{\rho}.
$$
Solving the two linear azimuthal invariants gives
$$
\boxed{u_\phi=
\frac{M_A^2\ell/r-r\omega}{M_A^2-1}},
$$
$$
\boxed{B_\phi=
\frac{\mu_0k(\ell/r-r\omega)}{M_A^2-1}}.
$$
At an <Alfvén surface>, $M_A^2=1$. Smooth passage through the apparent singularity requires both numerators to vanish at the same cylindrical radius $r_A$, giving the <Alfvén-surface regularity condition for an axisymmetric wind>
$$
\boxed{\ell=\omega r_A^2}.
$$
The finite limiting values of $u_\phi$ and $B_\phi$ then follow by l'Hopital's rule from the local variation of $M_A$ and $r$ along the field line; the algebraic invariants alone fix their combination rather than each value separately at the critical point.
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