Polynomial harmonic morphisms and eigenfamilies on spheres

The eigenfamilies of Gudmundsson and Sakovich can be used to generate harmonic morphisms, proper $r$-harmonic maps, and minimal co-dimension $2$ submanifolds. This article begins by characterising the globally defined eigenfamilies of the sphere $S^m$; they correspond to orthogonal families of homogeneous polynomial harmonic morphisms from $\Bbb{R}^{m+1}$ to $\Bbb C$, all of the same degree. We investigate and construct such families, paying special attention to those that are not congruent to families of holomorphic polynomials. Strong restrictions for families of such polynomials are found in low dimensions, and the pairs of degree $2$ maps that induce an eigenfamily are classified.