  The twistor space of the moduli space of solutions of Hitchin's self-duality
equations can be identified with the Deligne-Hitchin moduli space of
$\lambda$-connections. We use real projective structures on Riemann surfaces to
prove the existence of new components of real holomorphic sections of the
Deligne-Hitchin moduli space. Applying the twistorial construction we show the
existence of new hyper-K\"ahler manifolds associated to any compact Riemann
surface of genus $g\geq2$. These hyper-K\"ahler manifolds can be considered as
moduli spaces of (certain) singular solutions of the self-duality equations.
