  We prove that each measure $\mu$ in $R^4$ admits an equipartition by 4
hyperplanes, provided that it is symmetric with respect to a 2-dimensional,
affine subspace $L$ of $R^4$. Moreover we show, by computing the complete
obstruction in the relevant group of normal bordisms, that without the symmetry
condition, a naturally associated topological problem has a negative solution.
The computation is based on the Koschorke's exact singularity sequence and the
remarkable properties of the essentially unique, balanced binary Gray code in
dimension 4.
