  We consider the solution of the vector Dyson equation $-1/m=z+Sm$ in the case
when $S$ has a block staircase structure with $(n-1)$ different critical zero
blocks below the strictly positive anti-diagonal and all elements right above
the anti-diagonal are strictly positive. We prove that the components of $m$
behave as fractional powers of $z$ in the neighbourhood of zero and show that
the self-consistent density of states $\rho(E)$ behaves as $\vert
E\vert^{-\frac{n-1}{n+1}}$ as $E$ tends to zero, where $n^2$ is a number of
blocks. Both constant block and non-constant block cases are considered. In the
non-constant case uniform estimates for the components of $m$ are obtained.
