Analytic Classification of Fuchsian Singular Points

We study the local analytic classification of Fuchsian singular points. The resonance formal normal form (FNF) of a system with a Fuchsian singular point, as well as the local analytic equivalence of a system to its resonance FNF, is well known. However, there are distinct resonance FNFs locally analytically equivalent to each other. The main theorem of the paper reduces the problem of local analytic equivalence of resonance FNFs to a problem about conjugacy of certain matrices associated to two FNFs (which are nil-triangular) by a block upper triangular matrix. As a consequence, the local analytic classification of Fuchsian singular points reduces to the study of the orbits of the group of block upper triangular matrices on nil-triangular matrices by conjugation.