I have been reading this paper and it says (see the last paragraph in the screenshot below) that 'the latter condition requires that $B_\theta$ and $B_\phi$ vanish along the axis $\mu=1,\ -1$'. Why does $B_\theta$ and $B_\phi$ need to be zero at $\theta=0,\ \pi$?
I think the reason $B_\theta$ and $B_\phi$ must be zero at $\theta=0$ and $\theta=\pi$ is that the value of $\phi$ is ambiguous at these coordinates. Since the unit vectors are given by $$\mathbf{\hat{\theta}} = \cos(\theta)\cos(\phi)\mathbf{\hat{x}} + \cos(\theta)\sin(\phi)\mathbf{\hat{y}} - \sin(\theta)\mathbf{\hat{z}}$$ $$\mathbf{\hat{\phi}}=-\sin(\phi)\mathbf{\hat{x}} + \cos(\phi)\mathbf{\hat{y}},$$ if $B_\theta$ or $B_\phi$ are non-zero at $\theta=0$ or $\theta=\pi$ then the direction the vector points in is ambiguous. Hence, setting $B_\theta=B_\phi=0$ at $\theta=0$ and $\theta=\pi$ removes this ambiguity.