## When is the monodromy group of a linear differential equation dense in the Galois group?

Given a system $Y'=A(t)Y$ with only regular singular points, then a theorem of Schlesinger says that the Zariski closure of the monodromy group is equal to the Galois group of the corresponding Picard-Viessot extension. I know that there is a similar statement in the general case of rational differential equations or systems with irregular singular points, but instead of just the monodromy group one has to consider more analytic information, the stokes operators and the exponential torus.

This may be a silly question, but I wanted to ask if Schlesinger's theorem can be extended to some class of differential equations/systems with irregular singularities, but that are nice in some way. Or even better, a characterization, for which systems is the monodromy group dense in the differential Galois group?

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