On the Deligne-Simpson problem and its weak version

We consider the Deligne-Simpson problem (DSP) (resp. the weak DSP): Give necessary and sufficient conditions upon the choice of the p + 1 conjugacy classes cj ⊂ gl(n,C) or Cj ⊂ GL(n,C) so that there exist irreducible (p+ 1)-tuples (resp. (p+ 1)-tuples with trivial centralizers) of matrices Aj ∈ cj with zero sum or of matrices Mj ∈ Cj whose product is I. The matrices Aj (resp. Mj) are interpreted as matrices-residua of Fuchsian linear systems (resp. as monodromy matrices of regular linear systems) of differential equations with complex time. In the paper we give sufficient conditions for solvability of the DSP in the case when one of the matrices is with distinct eigenvalues.