An estimate for the average number of common zeros of Laplacian eigenfunctions

On a compact Riemannian manifold $ M$ of dimension $ n$, we consider $ n$ eigenfunctions of the Laplace operator $ \Delta $ with eigenvalue $ \lambda $. If $ M$ is homogeneous under a compact Lie group preserving the metric then we prove that the average number of common zeros of $ n$ eigenfunctions does not exceed $ c(n)\lambda ^{n/2}{\rm vol}\,M$, the expression known from the celebrated Weyl's law. Moreover, if the isotropy representation is irreducible, then the estimate turns into the equality. The constant $ c(n)$ is explicitly given. The method of proof is based on the application of Crofton's formula for the sphere.