= Polynomial Lax representation of the Nahm equations
{title2=$\dot L(\lambda)=[L(\lambda),M(\lambda)]$}
Put $P=A_1+iA_2$, $Q=A_1-iA_2$ and $R=A_3$, and define
$$
L(\lambda)=P+2R\lambda-Q\lambda^2,\qquad
M(\lambda)=-iR+iQ\lambda.
$$
Direct expansion gives $[L,M]=-i[P,R]+i[P,Q]\lambda+i[R,Q]\lambda^2$. The <Nahm equations> imply $\dot P=-i[P,R]$, $2\dot R=i[P,Q]$ and $-\dot Q=i[R,Q]$, so $\dot L=[L,M]$. In any finite-dimensional <matrix> representation, cyclicity of the <matrix trace> gives
$$
\frac d{dt}\operatorname{Tr}L^p
=p\operatorname{Tr}(L^{p-1}[L,M])=0\qquad(p=1,2,\ldots).
$$
Thus every coefficient of the degree-at-most-$2p$ <matrix trace> polynomial is conserved. At projective infinity the polynomial is interpreted as a section of $\mathcal O(2p)$, not a globally <holomorphic> scalar function on the <complex projective line>.
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