= Solution
All invariants in this part refer to the selected <magnetic field line>. Its squared poloidal <Alfvén number> is
$$
M_{Ap}^2=\frac{\mu_0\rho u_p^2}{B_p^2}=\frac{\mu_0k^2}{\rho}.
$$
At the <Alfvén surface> crossing it equals unity, so $\rho_a=\mu_0k^2$ and $M_{Ap}^2=1/y$. Subtract the two azimuthal integrals in part (a) to find
$$
R\omega-\frac\ell R=\frac{B_\phi}{\mu_0k}\left(1-\frac{\mu_0k^2}{\rho}\right),\qquad
v_\phi\equiv u_\phi-R\omega=\frac{R\omega-\ell/R}{y-1}.
$$
Finite azimuthal fields and <velocity> at $y=1$ require the <Alfvén-surface regularity condition for an axisymmetric wind> $\ell=\omega R_a^2$. Hence
$$
v_\phi=\omega R_a\frac{x-x^{-1}}{y-1},\qquad
|u_p|=\frac{|k|C}{\rho_aR_a^2x^2y}
=\frac{C}{\sqrt{\mu_0\rho_a}R_a^2x^2y}.
$$
Here $v_\phi$ denotes the azimuthal <velocity> relative to field-line rotation. For gravity of a <point mass> at $z=0$, $\Phi=-GM/R=-GM/(R_ax)$.
Insert these quantities in the corotating <energy> integral with $w=0$, and divide by the gravitational <energy> scale $GM/R_a$. This gives the <cold radial magnetohydrodynamic wind integral>,
$$
\boxed{f(x,y)=\frac\alpha2\left[\left(\frac{x-x^{-1}}{y-1}\right)^2-x^2\right]
+\frac\beta{2x^4y^2}-\frac1x=\frac{\varepsilon R_a}{GM}},
$$
with
$$
\boxed{\alpha=\frac{\omega^2R_a^3}{GM},\qquad
\beta=\frac{C^2}{\mu_0\rho_a GM R_a^3}}.
$$
Both are dimensionless. The apparent $0/0$ at $x=y=1$ must be evaluated on the smooth solution branch, not assigned an arbitrary value.
A constant-<energy> curve obeys $f_x+f_y\,dy/dx=0$. At a nondegenerate magnetosonic point, the vanishing coefficient of the <mass density> derivative requires simultaneous vanishing of its numerator:
$$
\boxed{f_y=0,\qquad f_x=0,\qquad f=f_{\rm wind}}.
$$
The curve must admit a real crossing slope. Expanding about such a point gives $f_{yy}m^2+2f_{xy}m+f_{xx}=0$, $m=dy/dx$, so a regular <saddle point of a scalar function> crossing has $f_{xy}^2-f_{xx}f_{yy}>0$. These are the form of the slow and fast regularity conditions in a warm wind, with the <enthalpy> contribution retained in $f$.
In the strictly cold approximation used to obtain the displayed $f$, the slow speed is zero. There is a <cold-limit degeneracy of a slow magnetosonic point>: a finite-speed interior slow crossing need not survive. Indeed,
$$
f_y=-\frac{\alpha(x-x^{-1})^2}{(y-1)^3}-\frac{\beta}{x^4y^3}<0\qquad(y>1,\ \beta>0),
$$
so this cold equation has no ordinary sub-Alfvénic slow critical point. A small retained <enthalpy> is needed to locate that point, which can approach the launching boundary as the <sound speed> tends to zero. The fast critical point still obeys the simultaneous conditions above wherever the reduced expression is nonsingular.
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