  Nonlinear perturbation of Fuchsian systems are studied in regions including
two singularities. Such systems are not necessarily analytically equivalent to
their linear part (they are not linearizable). Nevertheless, it is shown that
in the case when the linear part has commuting monodromy, and the eigenvalues
have positive real parts, there exists a unique correction function of the
nonlinear part so that the corrected system becomes analytically linearizable.
