Signatures of branched coverings and solvability in quadratures

The signature of a branched covering over the Riemann sphere is the set of its branching points together with the orders of local monodromy operators around them.
What can be said about the monodromy group of a branched covering if its signature is known? It seems at first that the answer is nothing or next to nothing. It turns out however that an elliptic signature determines the monodromy group completely and a parabolic signature determines it up to an abelian factor. For these non-hyperbolic signatures (with one exception) the corresponding monodromy groups turn out to be solvable.
The algebraic functions related to all (except one) of these signatures are expressible in radicals. As an example, the inverse of a Chebyshev polynomial is expressible in radicals. Another example of this kind is provided by functions related to division theorems for the argument of elliptic functions. Such functions play a central role in a work of Ritt which inspired this work.
Linear differential equations of Fuchs type related to these signatures are solvable in quadratures (in the case of elliptic signatures – in algebraic functions). A well-known example of this type is provided by Euler differential equations, which can be reduced to linear differential equations with constant coefficients.