Nahm's equations with poles and conservation of characteristic polynomial 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 $y$.
 A: I just stumbled upon this question, my apologies if you have already figured it out in the meantime.
The field $\mathcal{X}$ in this reference is an $\mathfrak{sl}_n\mathbb{C}$-valued function. Its characteristic polynomial is just the usual characteristic polynomial $p_\mathcal{X} = \det (\lambda \mathbf{1} - \mathcal{X})$ (evaluated pointwise).
The $y$-independence then follows from the complex Nahm pole boundary condition and the fact that the characteristic polynomial is an invariant polynomial.
A: Equation (3.7) tells you that $\mathcal{X}$ and $\mathcal{A}$ are an example of what is called a Lax pair in integrable systems.  The Wikipedia page for Lax pairs (which links off the one for Nahm's equations) explains why it follows that the eigenvalues of $\mathcal{X}$ are constant in $y$. 
If like me you like  bundles and connections, note that (3.7) says that the field (or matrix valued function)  $\mathcal{X}$ is a covariantly constant endomorphism of the trivial bundle with connection defined by the matrix valued function $\mathcal{A}$. If you solve the eigenvalue equation $\mathcal{X} v = \lambda v $ at one 
point $y_0$ and then extend $v$ to a covariantly constant section it will remain an eigenvector of $\mathcal{X}$ for all $y$. To see this calculate or note that $\mathcal{X} v - \lambda v $ is a covariantly constant section which vanishes at $y_0$ and thus is zero everywhere. So $\mathcal{X}(y)$ has eigenvalue $\lambda$ for all $y$.  Hence the usual characteristic polynomial of $\mathcal{X}(y)$, being made up of symmetric functions applied to eigenvalues is constant in $y$. 
