Alfvénic degeneracy of steady planar MHD ($B_z({\mu }_0{k}^2/\rho -1)={\mu }_{0}k(V-U)$)

The two axial invariants in steady planar ideal-MHD field-line invariants give the displayed identity. When $V-U\ne 0$, a finite smooth flow cannot reach horizontal Alfvén speed. When $V=U$, an everywhere Alfvénic branch is possible even with $B_z\ne 0$: constant $\rho$ and $\mathbf{u}=\mathbf{B}/\sqrt{{\mu }_0\rho }$ give an example. The nondegenerate exclusion must not be applied to this branch.