= Solution
The <modified Helmholtz closed spectral one-form> is exact in the simply connected quadrant. Writing its primitive as $\mu E$ gives the compatible first-order equations
$$
\mu_z-ik\mu=q_z+ikq,\qquad\mu_{\bar z}-\frac\lambda{ik}\mu=-q_{\bar z}-\frac\lambda{ik}q.
$$
Their compatibility is precisely the PDE already proved in part (a). Set $H=E^{-1}=e^{ia(k)x-b(k)y}$. Construct three primitives: $\mu_0$ starts at the corner, $\mu_x$ starts at the infinite end of a horizontal line through the point, and $\mu_y$ starts at the infinite end of a vertical line. More explicitly,
$$
\begin{aligned}
\mu_0(z,k)&=H(z,k)\int_0^z W,\\
\mu_x(x,y,k)&=-i\int_x^\infty e^{-ia(k)(s-x)}\left[b(k)q(s,y)-q_y(s,y)\right]ds,\\
\mu_y(x,y,k)&=-\int_y^\infty e^{b(k)(s-y)}\left[iq_x(x,s)-a(k)q(x,s)\right]ds.
\end{aligned}
$$
Path independence follows from closedness. The two infinite primitives exist in the lower and left spectral half-planes respectively. In their common third quadrant they coincide, because the <global relation> makes the difference of the two basepoint integrals zero.
Choose a sectionally analytic spectral function by using $\mu_0$ in quadrant I, $\mu_y$ in quadrant II and $\mu_x$ in quadrants III and IV. It has no jump across the negative axes. Let $\ell_x=(0,\infty)$ and $\ell_y=(0,i\infty)$, both oriented outward. With plus denoting the left side of an oriented ray, the two remaining jumps are
$$
\mu_+-\mu_-=H\rho_x\quad\hbox{on }\ell_x,\qquad\mu_+-\mu_-=H\rho_y\quad\hbox{on }\ell_y.
$$
For example, on $\ell_x$ the jump is $\mu_0-\mu_x$, the primitive around the bottom boundary; on $\ell_y$ it is $\mu_y-\mu_0$, the left boundary traversed downward. This fixes both orientations and signs of the <Riemann-Hilbert problem>.
Integration by parts in the infinite primitive formulas, or in the finite primitive with its decaying basepoint contribution, gives
$$
\mu=-q+O(k^{-1})\quad(k\to\infty),\qquad\mu=q+O(k)\quad(k\to0).
$$
For instance the horizontal formula has leading value $-bq/a$, which tends to $-q$ at infinity and $q$ at zero; the vertical formula has leading value $-aq/b$ with the same limits. These expansions hold in closed subsectors, with the boundary limits obtained by oscillatory regularization. Along $\ell_x$, $H$ damps with $y>0$ at both spectral ends; along $\ell_y$, it damps with $x>0$. Thus the ray integrals converge in the interior and small and large spectral circles can be removed.
Solve the additive <Riemann-Hilbert problem> by a <Cauchy integral formula>, normalized at infinity:
$$
\mu(k)=-q+\frac1{2\pi i}\left[\int_{\ell_x}\frac{H(s)\rho_x(s)}{s-k}ds+\int_{\ell_y}\frac{H(s)\rho_y(s)}{s-k}ds\right].
$$
The <Sokhotski–Plemelj theorem> gives the stated jumps. The difference between this expression and the constructed primitive is analytic across both rays, bounded at zero and vanishes at infinity; the <Liouville theorem> makes that difference zero. Evaluate at zero, where $\mu(0)=q$. The difference between the two spectral normalizations is $2q$, hence the <quadrant modified Helmholtz spectral reconstruction> is
$$
\boxed{q(x,y)=\frac1{4\pi i}\left[\int_0^\infty e^{ia(k)x-b(k)y}\rho_x(k)\frac{dk}{k}+\int_0^{i\infty}e^{ia(k)x-b(k)y}\rho_y(k)\frac{dk}{k}\right].}
$$
The factor $1/(4\pi i)$ is essential. A useful independent normalization check is to write $p=a(k)$ on the real ray, so $b=\sqrt{p^2+4\lambda}=B(p)$ and $dk/k=dp/B$. On the imaginary ray write $k=ir$, $p=r-\lambda/r$ and $a=iB(p)$. The formula becomes
$$
\begin{aligned}
q(x,y)={}&\frac1{4\pi}\int_{\mathbb R}e^{ipx-B(p)y}\left[\widehat h_2(p)-\frac{\widehat n_2(p)}{B(p)}\right]dp\\
&+\frac1{4\pi}\int_{\mathbb R}e^{-B(p)x-ipy}\left[\widehat h_1(-p)-\frac{\widehat n_1(-p)}{B(p)}\right]dp.
\end{aligned}
$$
Each hat here is a half-line Fourier transform. This is exactly the spectral form of <Green second identity> with the decaying full-plane <fundamental solution>. The representation still contains both value and derivative traces; the next part eliminates the latter.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/iii/paper-86-spectral-contours.png]
{title=Oriented half-line drift contour and quadrant modified Helmholtz reconstruction rays with their analytic and global-relation domains}
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