  We prove in two different ways that the monodromy map from the space of
irreducible $\mathfrak{sl}_2$-differential-systems on genus two Riemann
surfaces, towards the character variety of $\mathrm{SL}_2$-representations of
the fundamental group, is a local diffeomorphism. This is motivated by a
question raised by \'Etienne Ghys about Margulis' problem: existence of curves
of negative Euler characteristic in compact quotients of
$\mathrm{SL}_2(\mathbb{C})$.
