# Lax Pairs for Linear PDEs

I'm trying to understand the discussion around equation (2.1) of the paper http://www.jstor.org/stable/53053. It says that the linear PDE $M(\partial_x,\partial_y)q=0$ with constant coefficients has the Lax pair $\mu_x+ik\mu=q$ and $M(\partial_x,\partial_y)\mu=0$, where k is any complex number and $\mu$ is a function.

The way I'm used to thinking of Lax pairs is as operators $L$ and $B$ such that $\dot{L}+[L,B]=0$ is equivalent to the original PDE. This is equivalent to requiring that the equations $L\phi=\lambda\phi$ and $\dot{\phi}=B\phi$ are compatible, where $\lambda$ is a fixed eigenvalue and $\phi$ is any function. Can anyone explain how this connects with the discussion in the paper? What are $L$ and $B$ in the above case?

Thanks!

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