Completely reducible factors of harmonic polynomials of three variables

We describe the divisors of complex valued homogeneous harmonic polynomials on $\mathbb R^{3}$ which are products of linear forms and characterize the homogeneous polynomials $p$ that admit a couple of linear forms $\ell_{1}$ and $\ell_{2}$ such that $\ell_{1}^{m}p$ and $\ell_{2}^{m}p$ are harmonic for some $m\in\mathbb N$. The latter gives an example of a pair of spherical harmonics whose set of common zeros has length that is compatible with the upper bound of this quantity for a single harmonic.