= Solution
Use the <linear dispersive Stokes equation> with time variable $t$. Put $E_k=e^{-ikx+\omega(k)t}$. Expanding the required <local relation> gives
$$
(E_kq)_t+(E_kX)_x=E_k\bigl(q_t+\omega q+X_x-ikX\bigr).
$$
Choose $X=q_{xx}+ikq_x+(1-k^2)q$. Direct differentiation then yields
$$
X_x-ikX=q_{xxx}+q_x-ik(1-k^2)q.
$$
The unwanted coefficient of $q$ cancels precisely when $\omega(k)=ik(1-k^2)$. Hence
$$
\boxed{\omega(k)=i(k-k^3),\qquad X=q_{xx}+ikq_x+(1-k^2)q.}
$$
With these choices, the <local relation> is exactly $E_k(q_t+q_x+q_{xxx})=0$ for every complex <spectral parameter for a linear boundary value problem>.
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