  We consider conformal immersions of Riemann surfaces in $\bb{S}^4$ and study
their Gauss maps with values in the Grassmann bundle $\mathcal{F} = SO_5/T^2
\to \mathbb{S}^4$. The energy of maps from Riemann surfaces into $\mathcal{F}$
is considered with respect to the normal metric on the target and immersions
with harmonic Gauss maps are characterized. We also show that the
normal-harmonic map equation for Gauss maps is a completely integrable system,
thus giving a partial answer of a question posed by Y. Ohnita in \cite{ohnita}.
Associated $\mathbb{S}^1$-families of parallel mean curvature immersions in
$\mathbb{S}^4$ are considered. A lower bound of the normal energy of Gauss maps
is obtained in terms of the genus of the surface.
