= Solution
Differentiating the first auxiliary equation in $t$, the second in $x$, and equating $\psi_{xt}$ and $\psi_{tx}$ leaves precisely
$$
q_t+q_{xxx}=0,
$$
so the <Airy equation> is their <compatibility condition for an overdetermined linear system>[compatibility condition].
Multiplication of $\psi_x-ik\psi=q$ by $e^{-ikx}$ gives
$$
(e^{-ikx}\psi)_x=e^{-ikx}q.
$$
Hence
$$
\psi_+(x,t,k)=e^{ikx}\int_{-\infty}^{x}e^{-iky}q(y,t)\,dy.
$$
For $\operatorname{Im}k\geq0$, the kernel $e^{ik(x-y)}$ is bounded when $y\leq x$; dominated differentiation makes $\psi_+$ analytic for $\operatorname{Im}k>0$. Rapid decrease gives
$$
\lim_{x\to+\infty}e^{-ikx}\psi_+(x,t,k)=\widehat q(k,t).
$$
For real $k$, multiply the time equation by $e^{-ikx}$ and let $x\to+\infty$. The rapidly decreasing terms on its right vanish, leaving
$$
\widehat q_t-ik^3\widehat q=0.
$$
Therefore
$$
\widehat q(k,t)=e^{ik^3t}\widehat q(k,0),
$$
and the <Fourier inversion theorem> gives
$$
\boxed{q(x,t)=\frac1{2\pi}\int_{\mathbb R}
e^{i(kx+k^3t)}\widehat q(k,0)\,dk}.
$$
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