  We deal here with the geometry of the twistor fibration $\mathcal{Z} \to
\bb{S}^3_1$ over the De Sitter 3-space. The total space $\mathcal{Z}$ is a five
dimensional reductive homogeneous space with two canonical invariant almost CR
structures. Fixed the normal metric on $\mathcal{Z}$ we study the harmonic map
equation for smooth maps of Riemann surfaces into $\mathcal{Z}$. A
characterization of spacelike surfaces with harmonic twistor lifts to
$\mathcal{Z}$ is obtained. It is also shown that the harmonic map equation for
twistor lifts can be formulated as the curvature vanishing of an
$\bb{S}^1$-loop of connections i.e. harmonic twistor lifts exist within
$\bb{S}^1$-families. Special harmonic maps such as holomorphic twistor lifts
are also considered and some remarks concerning (compact) vacua of the twistor
energy are given.
