  A new proof of a theorem that describes $(\lambda, \mu)$-eigenfunctions on
sphere is obtained. This proof is based on a statement that a function $f$ is a
$(\lambda, \mu)$-eigenfunction if and only if $f$ and $f^2$ are eigenfunctions
for Laplace-Beltrami operator. Also, minimal submanifolds in spheres of
codimension 1 are constructed. In the case of $\mathbb{S}^3$ two immersions of
Clifford torus and all Lawson $\tau_{n, m}$ surfaces are described in terms of
$(\lambda, \mu)$-eigenfunctions.
