  Let G be a piecewise constant $n\times n$ matrix function which is defined on
a smooth closed curve $\Gamma$ in the complex sphere and which has m jumps. We
consider the problem of determining the partial indices of the factorization of
the matrix function G in the space $L^p(\Gamma)$. We show that this problem can
be reduced to a certain problem for systems of linear differential equations.
  Studying this related problem, we obtain some results for the partial indices
for general n and m. A complete answer is given for n=2, m=4 and for n=m=3. One
has to distinguish several cases. In some of these cases, the partial indices
can be determined explicitly. In the remaining cases, one is led to two
possibilities for the partial indices. The problem of deciding which is the
correct possibility is equivalent to the description of the monodromy of n-th
order linear Fuchsian differential equations with m singular points.
