  We consider the weak version of the Deligne-Simpson problem: give necessary
and sufficient conditions upon the conjugacy classes $c_j\subset gl(n,{\bf C})$
(resp. $C_j\subset GL(n,{\bf C})$) so that there exist $(p+1)$-tuples of
matrices $A_j\in c_j$, $A_1+... +A_{p+1}=0$ (resp. $M_1... M_{p+1}=I$) with
trivial centralizers (i.e. reduced to scalars). The true Deligne-Simpson
problem requires irreducibility instead of triviality of the centralizer. When
the eigenvalues are generic, a Criterium on the Jordan normal forms defined by
the conjugacy classes gives the necessary and sufficient conditions for
solvability of the true problem. For index of rigidity 2 (i.e. when the sum of
the dimensions of the conjugacy classes equals $2n^2-2$) we show that for a
sufficiently large class of $(p+1)$-tuples of conjugacy classes the answer to
the weak problem is negative. These conjugacy classes define Jordan normal
forms that satisfy the Criterium.
