A generalisation of the Hopf Construction and harmonic morphisms into ${\text{s}}^2$

In this paper we construct a new family of harmonic morphisms $\phi :V^5\to {\text{s}}^2$, where $V^5$ is a 5-dimensional open manifold contained in an ellipsoidal hypersurface of \c^4=\r^8. These harmonic morphisms admit a continuous extension to the completion ${V^{\ast }}^5$, which turns out to be an explicit real algebraic variety. We work in the context of a generalization of the Hopf construction and equivariant theory.