  We consider extensions of the Rattray theorem and two Makeev's theorems,
showing that they hold for several maps, measures, or functions simultaneously,
when we consider orthonormal $k$-frames in $\R^n$ instead of orthonormal basis
(full frames).
  We also present new results on simultaneous partition of several measures
into parts by $k$ mutually orthogonal hyperplanes.
  In the case $k=2$ we relate the Rattray and Makeev type results with the well
known embedding problem for projective spaces.
