= Solution
Let $E=e^{-ikx+k^2t}$. Expanding $\partial_t(Eq)+\partial_x(EX)$ shows that the required flux is
$$
\boxed{X=-q_x-ikq.}
$$
Indeed $X_x-ikX=-q_{xx}-k^2q$, cancelling the $k^2q$ from the time <derivative>. Introduce a potential $\varphi$ by $\varphi_x=Eq$, $\varphi_t=E(q_x+ikq)$. Equality of its mixed <derivatives> is precisely the conservation law. With $\varphi=E\mu$, these equations become the linear auxiliary system
$$
\boxed{\mu_x-ik\mu=q,\qquad\mu_t+k^2\mu=q_x+ikq.}
$$
This is a <Lax pair> in the affine-potential form used for linear evolution equations. If a homogeneous <matrix> pair is preferred, set $\Psi=(\mu,1)^T$ and use $\Psi_x=\left(\begin{smallmatrix}ik&q\\0&0\end{smallmatrix}\right)\Psi$, $\Psi_t=\left(\begin{smallmatrix}-k^2&q_x+ikq\\0&0\end{smallmatrix}\right)\Psi$. Its <zero-curvature condition> has upper-right entry $q_t-q_{xx}$ and all other entries zero.
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