  We study a natural functional on the space of holomorphic sections of the
Deligne-Hitchin moduli space of a compact Riemann surface, generalizing the
energy of equivariant harmonic maps corresponding to twistor lines. We give a
link to a natural meromorphic connection on the hyperholomorphic line bundle
recently constructed by Hitchin. Moreover, we prove that for a certain class of
real holomorphic sections of the Deligne-Hitchin moduli space, the functional
is basically given by the Willmore energy of corresponding (equivariant)
conformal map to the 3-sphere. As an application we use the functional to
distinguish new components of real holomorphic sections of the Deligne-Hitchin
moduli space from the space of twistor lines.
