[Proj] specific projection on the surface of WGS84 ellipsoid

[Aleksander Yanovskiy]

Greetings!

Could you please help me with the following task. It is very specific, so any suggestion is welcome.
I need to divide the surface of the WGS84 ellipsoid into several sectors by lines which connect the North and South Magnetic Poles and which are situated on flat planes, passing through North-South magnetic axis at equal angles between adjacent planes. (The same in other words: we take some Earth ellipsoid (say WGS84), locate the North and South Magnetic Poles on it's surface (they are not antipodal), connect the poles by straight line, divide the space around the line into several equiangular sectors. Where do the planes between the sectors cross the surface of our ellipsoid?)

Sincerely,
Aleksander