 Solution

Let $\Phi$ denote the common trace transform of the unique solution, or of the rotational average at an exceptional parameter. Because $q$ and $f$ are real and $\lambda$ is real, $\overline{\Phi (\bar{k})}=\Phi (k)$ and $\overline{\Psi (\bar{k})}=\Psi (k)$. Taking the Schwarz conjugate of the global relation means replacing $k$ by $\bar{k}$ and conjugating the whole equation. It yields the second of the conjugate global relations for the modified Helmholtz equation:In this conjugation the rotated arguments interchange; after putting terms in the order $k,\bar{a}k,ak$, the coefficients have the displayed signs and arguments. DefineThe elementary cube-root identities giveMultiply the first global relation by $E(iak)$ and the conjugate one by $E(-iak)$. Then the coefficient of $\Phi (ak)$ is one in both equations:where all the quantities on the right are known,Subtracting eliminates $\Phi (ak)$. Ifthe resulting solution for the requested transform isAt a zero of the denominator this identity is interpreted in its undivided form. The transforms themselves remain finite for $k\ne 0$; consequently the numerator must vanish. That cancellation, rather than evaluation of a singular quotient, determines the Fourier coefficients below.