Factorization of special harmonic polynomials of three variables

We consider homogeneous 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 homogeneous rotation invariant polynomials $p$ of degree $m$ such that $p$ is reducible over the rationals and $(x+yi)^{n}p$ is harmonic for some $n\in\mathbb{N}$. We describe $\mathfrak{R}_{m}$ for $m\leq5$ and prove that $\mathfrak{R}_{6}$ and $\mathfrak{R}_{7}$ are finite.