On the Deligne-Simpson problem and its weak version

We consider the Deligne-Simpson problem (DSP) (respectively the weak DSP): Give necessary and sufficient conditions upon the choice of the p+1 conjugacy classes or so that there exist irreducible (p+1)-tuples (respectively (p+1)-tuples with trivial centralizers) of matrices A_j∈c_j with zero sum or of matrices M_j∈C_j whose product is I. The matrices A_j (respectively M_j) are interpreted as matrices-residua of Fuchsian linear systems (respectively 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.