In book "Introduction to Classical Integrable Systems". The proposition in p.p. 36 writes "Assuming that $L(\lambda)$ has distinct eigenvalues in a neighbourhood of $\lambda_k$,..." then $L(\lambda)$ allows for a good diagonalization. (You can also look at p.p. 52 of slides https://www.math.uni-hamburg.de/home/alim/LaxPairsZakharovShabat.pdf for something similar)
The condition looks strange to me, say consider $\begin{pmatrix}0&1\\1/\lambda&0 \end{pmatrix}$ near $\lambda=0$. They has eigenvalues $\pm \sqrt{1/\lambda}$, which is distinct unless $\lambda=\infty$. But seems that's not the matrix we look for, I guess the correct condition should be that the coefficients of highest pole $L_{k,-n_r}$ has distinct eigenvalues. Am I correct?