  In this paper we consider special linear Fuchsian systems of rank $2$ on a
$4-$punctured sphere and the corresponding parabolic structures. Through an
explicit abelianization procedure we obtain a $2-$to$-1$ correspondence between
flat line bundle connections on a torus and these Fuchsian systems. This
naturally equips the moduli space of flat $SL(2,\mathbb C)-$connections on a
$4-$punctured sphere with a new set of Darboux coordinates. Furthermore, we
apply our theory to give a complex analytic proof of Witten's formula for the
symplectic volume of the moduli space of unitary flat connections on the
$4-$punctured sphere.
