It seems that this problem very simple.
Let F=[v1,F1]+...+[vN,FN]- sum of  moments of forces,
where vi=(cos(fi)cos(hi),cos(fi)sin(fi),sin(fi))- vector
of point and Fi=(cos(qi)cos(zi),cos(qi)sin(zi),sin(qi))-
vector of force Fi.
Let V,G- point and force to be found.
Then we have [V,G]=-F,|G|->min;
We must have V- is normal to F=>V any point on meridian
and G=-[V,F]-calculated after
Thus solution ambiguous  even with demand |G|->min

And why do you think that this problem must have an unique answer? Any correct answer will be OK.

Because when many solutions how  can we check
that difference with test's answer <=1E-8

This problem has special checking program. It doesn't use the difference between your answer and jury's one.

Everything needed to solve this problem is told in the first post!
And there is only simple 3D-geometry.