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The resulting positions of those points don´t match. The difference is about 5 percent, hence not trivial.

Its not a problem of the computation, several different approaches give the same result (Integrals, sums, one or three spatial dimensions, high working precision).

I had assumed those points to match and in the literature I found no obvious reason why they should not.

Why this difference?

A Mathematica program of a sample calculation can be given.