Seeing this statement in this answer
There are at least two spots on Earth where the magnetic field of the planet is not horizontal, but is vertical.
has got me thinking. I don't mean to quote it out of context, I'm using it as a spring board for a different situation.
For a realistic spherical harmonic model of the Earth's field (e.g. World Magnetic Model of degree and order 12), is it a mathematical necessity that there be two points on an spherical approximation of the Earth's surface where the direction of the field is normal to the sphere?
What about if an ellipsoidal surface was chosen to better reflect the shape of the Earth's surface? Would there still be at least two points that were normal?
note: I'm not using the word "vertical" in the expanded statement of the question as it would then require a second model of the Geopotential, and then different people would choose differently if a term reflecting a pseudo potential reflecting rotation should be included or not in the definition of "vertical".
below: "Geodynamo Between Reversals", from here to illustrate that the source of the Earth's surface field is something that should not be thought of as a simple dipole.