Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 59 / 1 / a / Solution

At both points, $\cos \theta =0$ and $\sin 2\varphi =0$, so both components of the vector field vanish. Thus both points are equilibrium points for every parameter value. These are regular points of the spherical coordinates chart: the coordinate singularity at the pole plays no role in their linearization.