Fuchsian System

A system of linear differential equations

$${dy\over dz}=A(z)y,$$

with $A(z)$ an Analytic $n\times n$ Matrix, for which the Matrix $A(z)$ is Analytic in $\bar{\Bbb{C}}\backslash\{a_1, \ldots, a_N\}$ and has a Pole of order 1 at $a_j$ for $j=1$, ..., $N$. A system is Fuchsian Iff there exist $n\times n$ matrices $B_1$, ..., $B_N$ with entries in $\Bbb{Z}$ such that

$$A(z)=\sum_{j=1}^N {B_j\over z-a_j}$$

$$\sum_{j=1}^N B_j=v.$$