  We consider the eigenfunctions of the Laplace operator $\Delta $ on a compact
Riemannian manifold of dimension $n$. For $M$ homogeneous with irreducible
isotropy representation and for a fixed eigenvalue of $\Delta $ we find the
average number of common zeros of $n$ eigenfunctions. For this we compute the
volume of the image of $M$ under an equivariant immersion into a sphere.
