Geodesic
Constructive Solvability Conditions for the Riemann--Hilbert Problem
Matematičeskie zametki, Tome 77 (2005) no. 5, pp. 643-655

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Sufficient and necessary conditions for the solvability of the Riemann–Hilbert problem are studied. These conditions consist in the possibility of constructing stable and semistable pairs (of bundles and connections) for a given monodromy. The obtained results make it possible to develop algorithms for testing the solvability conditions for the Riemann–Hilbert problem. 
@article{MZM_2005_77_5_a0,
     author = {I. V. Vyugin},
     title = {Constructive {Solvability} {Conditions} for the {Riemann--Hilbert} {Problem}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {643--655},
     publisher = {mathdoc},
     volume = {77},
     number = {5},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2005_77_5_a0/}
}
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I. V. Vyugin. Constructive Solvability Conditions for the Riemann--Hilbert Problem. Matematičeskie zametki, Tome 77 (2005) no. 5, pp. 643-655. http://geodesic.mathdoc.fr/item/MZM_2005_77_5_a0/