Let (M = G/H;g)denote a four-dimensional pseudo-Riemannian generalized symmetric space and g = m + h the corresponding decomposition of the Lie algebra g of G. We completely determine the harmonicity properties of vector fields belonging to m. In some cases, all these vector fields are critical points for the energy functional restricted to vector fields. Vector fields defining harmonic maps are also classified, and the energy of these vector fields is explicitly calculated.

Given a Hörmander system $X=\{X_1,\cdots ,X_m\}$ on a domain $\Omega \subseteq {𝐑}^n$ we show that any subelliptic harmonic morphism $\phi$ from $\Omega$ into a $\nu$-dimensional riemannian manifold $N$ is a (smooth) subelliptic harmonic map (in the sense of J. Jost & C-J. Xu, [9]). Also $\phi$ is a submersion provided that $\nu \le m$ and $X$ has rank $m$. If $\Omega ={𝐇}_n$ (the Heisenberg group) and $X=\left\{\frac{1}{2}\left(L_{\alpha }+L_{\overline{\alpha }}\right),\frac{1}{2i}\left(L_{\alpha }-L_{\overline{\alpha }}\right)\right\}$, where $L_{\overline{\alpha }}=\partial /\partial {\overline{z}}^{\alpha }-i{z}^{\alpha }\partial /\partial t$ is the Lewy operator, then a smooth map $\phi :\Omega \to N$ is a subelliptic harmonic morphism if and only if $\phi \circ \pi :(C\left({𝐇}_n\right),F_{{\theta }_0})\to N$ is a harmonic morphism, where $S^1\to C\left({𝐇}_n\right)\stackrel{\pi }{\to }\to {𝐇}_n$ is the canonical circle bundle and $F_{{\theta }_0}$ is the Fefferman...