つづき

Indeed Andrey A. Bolibrukh (1990) found a counterexample to Plemelj's statement.
This is commonly viewed as providing a counterexample to the precise question Hilbert had in mind;
Bolibrukh showed that for a given pole configuration certain monodromy groups can be realised by regular, but not by Fuchsian systems.
(In 1990 he published the thorough study of the case of regular systems of size 3 exhibiting all situations when such counterexamples exists.
In 1978 Dekkers had shown that for systems of size 2 Plemelj's claim is true.
Andrey A. Bolibrukh (1992) and independently Vladimir Kostov (1992) showed that for any size, an irreducible monodromy group can be realised by a Fuchsian system.
The codimension of the variety of monodromy groups of regular systems of size n with p+1 poles which cannot be realised by Fuchsian systems equals 2(n-1)p (Vladimir Kostov (1992)).)
Parallel to this the Grothendieck school of algebraic geometry had become interested in questions of 'integrable connections on algebraic varieties', generalising the theory of linear differential equations on Riemann surfaces.
Pierre Deligne proved a precise Riemann?Hilbert correspondence in this general context (a major point being to say what 'Fuchsian' means).
With work by Helmut Rohrl, the case in one complex dimension was again covered.
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