  The set of primitive vectors on large spheres in the euclidean space of
dimension d>2 equidistribute when projected on the unit sphere. We consider
here a refinement of this problem concerning the direction of the vector
together with the shape of the lattice in its orthogonal complement. Using
unipotent dynamics we obtained the desired equidistribution result in dimension
d>5 and in dimension d=4,5 under a mild congruence condition on the square of
the radius. The case of d=3 is considered in a separate paper.
