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Differential systems with a Fuchsian linear part are studied in regionsincluding all the singularities in the complex plane of these equations. Suchsystems are not necessarily analytically equivalent to their linear part (theyare not linearizable) and obstructions are found as a unique nonlinearcorrection after which the system becomes formally linearizable.  More generally, normal forms are found.  The corrections and the normal forms are found constructively. Expansions inmultiple orthogonal polynomials and their generalization to matrix-valuedpolynomials are instrumental to these constructions.

Differential systems with Fuchsian linear part: correction and linearization, normal forms and multiple orthogonal polynomials