= Solution
For the <Airy equation>, let $P=e^{-ikx+\omega t}q$. Its <local relation> follows by arranging the flux so that the unwanted spatial derivatives cancel:
$$
(\partial_x-ik)(-q_{xx}-ikq_x+k^2q)=-q_{xxx}-ik^3q.
$$
Consequently
$$
P_t-(e^{-ikx+\omega t}X)_x=e^{-ikx+\omega t}[q_t+q_{xxx}+(\omega+ik^3)q].
$$
Thus the <dispersion relation> and the required flux are \b[given by]
$$
\boxed{\omega(k)=-ik^3,\qquad X=-q_{xx}-ikq_x+k^2q.}
$$
The calculation is polynomial in the complex <spectral parameter>, so the <local relation> holds for every $k\in\mathbb C$.
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