= Solution
For a steady flow, <mass conservation> gives $\rho w=\text{constant}$ and $\rho'/\rho=-w'/w$. Define the <Alfvén velocity> $\mathbf v_a=\mathbf B/\sqrt{\mu_0\rho}$ and $C=w^2-v_{az}^2$. Combining the horizontal momentum and <MHD induction equations> without dividing by $C$ gives
$$
C B_x'=-wB_xw',\qquad
C B_y'=-wB_yw'+a(wB_x-B_zv_x).
$$
Dot these identities with $B_x/(\mu_0\rho)$ and $B_y/(\mu_0\rho)$, respectively:
$$
\frac{C}{\mu_0\rho}(B_xB_x'+B_yB_y')
=-w(v_{ax}^2+v_{ay}^2)w'
+a v_{ay}(wv_{ax}-v_{az}v_x).
$$
The vertical momentum equation and the <isothermal equation of state> give
$$
\left(w-\frac{c_s^2}{w}\right)w'
=-g-\frac{B_xB_x'+B_yB_y'}{\mu_0\rho}.
$$
Multiply by $C$ and eliminate the magnetic derivative to obtain
$$
\boxed{\left[w^4-(c_s^2+v_a^2)w^2+c_s^2v_{az}^2\right]\frac{w'}w
=g(v_{az}^2-w^2)+a v_{ay}(v_xv_{az}-wv_{ax}).}
$$
No division by $w^2-v_{az}^2$ was needed in deriving this necessary relation.
The <magnetosonic critical speeds> in the $z$ direction are
$$
c_{\mathrm f,\mathrm s}^2=
\frac12\left[c_s^2+v_a^2
\ \mathbin{\pm}\
\sqrt{(c_s^2+v_a^2)^2-4c_s^2v_{az}^2}\right].
$$
The plus sign gives the <fast magnetosonic wave> speed and the minus sign the <slow magnetosonic wave> speed. Thus the differential coefficient is $(w^2-c_{\mathrm f}^2)(w^2-c_{\mathrm s}^2)$. A smooth outflow proceeding from below both speeds to above both must normally pass through both <magnetosonic critical speeds>. At each crossing, \b[the right-hand side must also vanish]; this is the <regularity at a magnetosonic point> condition. The derivative coefficient changes sign at each nondegenerate crossing. For $w>0$ and $w'>0$, the driving term must be positive below the slow point, negative between the points and positive above the fast point.
The <Alfvén speed> component satisfies $c_{\mathrm s}^2\leq v_{az}^2\leq c_{\mathrm f}^2$. With downward gravity $g>0$, the gravitational term is consequently nonnegative at the slow point and nonpositive at the fast point. The <magnetohydrodynamic shear work> contribution must balance it at each point, and can provide the upward driving needed to pass the fast point. For $a=0$ and strictly positive $g$, generic separated slow and fast points cannot satisfy the required zero numerator; special vanishing-gravity or coincident-speed cases need separate treatment.
There is also <Alfvén-point compatibility in a plane-parallel sheared flow>. At $w^2=v_{az}^2$, the original transverse equations require
$$
wB_xw'=0,\qquad
wB_yw'=a(wB_x-B_zv_x).
$$
These restrictions are not generally visible as a zero of the scalar differential coefficient, which there equals $-v_{az}^2(v_{ax}^2+v_{ay}^2)$. In particular, a strictly accelerating regular solution must have $B_x=0$ at that point. The scalar relation is therefore a necessary wind equation, not a substitute for regularity of all the original <ideal magnetohydrodynamic equations>. Degenerate cases such as a purely longitudinal <magnetic field> can merge characteristic speeds and reduce the number of distinct critical conditions.
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