Alfvén-point compatibility in a plane-parallel sheared flow

At $w^2=v_{az}^2$, the transverse steady equations require $w{B}_x{w}^{\prime }=0$ and $w{B}_y{w}^{\prime }=a(w{B}_x-B_z{v}_x)$. These conditions can be lost by division when eliminating transverse magnetic derivatives. A scalar equation displaying the magnetosonic critical speeds need not display every compatibility condition at an Alfvén speed.