In more modern language, the (systems of) differential equations in question are those defined in the complex plane, less a few points, and with a regular singularity at those.
A few years later the Soviet mathematician Yuliy S. Il'yashenko and others started raising doubts about Plemelj's work.
In fact, Plemelj correctly proves that any monodromy group can be realised by a regular linear system which is Fuchsian at all but one of the singular points.
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.
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.
Pierre Deligne proved a precise Riemann–Hilbert correspondence in this general context (a major point being to say what 'Fuchsian' means).