In "Supersymmetric Boundary Conditions in N=4 Super Yang-Mills Theory" Nahm's equations are studied in Section 3. In particular, it is explained that their moduli space of solutions, $\mathcal{M}$, is a hyperkaehler manifold, and on page 33 it is described as a complex manifold in one of the complex structures. Here it is defined by the holomorphic equation (3.7), i.e., \begin{equation} \frac{d\mathcal{X}}{dy}+[\mathcal{A},\mathcal{X}]=\frac{\mathcal{D}\mathcal{X}}{\mathcal{D}y}=0, \end{equation} where $\mathcal{X}$ and $\mathcal{A}$ are complex coordinates (in a particular complex structure) on the space $\mathcal{W}$, whose hyperkaehler quotient is $\mathcal{M}$, and where $y\in [0,\infty]$. The equation above is equivalent to two of the Nahm's equations, and $\mathcal{M}$ can be obtained by dividing out the complexification of the relevant Lie group $G$.
In Section 3.3, solutions of this equation with poles at $y=0$ are studied, and in the last paragraph of page 41, it is noted that the characteristic polynomial of $\mathcal{X}$ is independent of $y$. My question is, what is the definition of the characteristic polynomial here, and how does the complex Nahm equation imply that it is independent of $y$?
My best guess is that they define the characteristic polynomial as $\textrm{det}(y\textbf{1}-\mathcal{X}(y))$, but I am unable to understand why this is constant along $y$.