  We consider harmonic polynomials of real variables $x,y,z$ that are
eigenfunctions of the rotations about the axis $z$. They have the form $(x\pm
yi)^{n}p(x,y,z)$, where $p$ is a rotation invariant polynomial. Let ${\mathfrak
R}_{m}$ be the family of the polynomials $p$ of degree $m$ which are reducible
over the rationals. We describe ${\mathfrak R}_{m}$ for $m\leq5$ and prove that
${\mathfrak R}_{6}$ and ${\mathfrak R}_{7}$ are finite.
