The Riemann-Hilbert mapping for ₂ -systems over genus two curves

We prove in two different ways that the monodromy map from the space of ₂-differential-systems on genus two Riemann, towards the character variety of ₂-representations of fundamental group, is a local diffeomorphism. This is motivated by a raised by \\'Etienne Ghys about Margulis' problem: existence of curves negative Euler characteristic in compact quotients of₂().