  Nonlinear perturbation of Fuchsian systems are studied in a region including
two singularities. It is proved that such systems are generally not
analytically equivalent to their linear part (they are not linearizable) and
the obstructions are found constructively, as a countable set of numbers.
Furthermore, assuming a polynomial character of the nonlinear part, it is shown
that there exists a unique formal "correction" of the nonlinear part so that
the "corrected" system is formally linearizable.
  Normal forms of these systems are found, providing also their classification.
