= Solution
Identical oriented <Dirichlet boundary data> imply a rotation-invariant solution whenever the Dirichlet problem is unique. For $\lambda\geq0$, uniqueness follows directly: the difference $h$ of two solutions has zero boundary trace, and <integration by parts> gives $\int_D|\nabla h|^2+4\lambda\int_D|h|^2=0$. Thus $h=0$. The same conclusion holds for a nonresonant negative parameter by invertibility of the <Dirichlet Laplacian> shift. Then $q(az)=q(z)$, all three normal traces equal a common $q_N(s)$, and $\Psi_j=\Psi$.
Set
$$
e(k)=e^{\ell(k+\lambda/k)/2},\qquad\Phi(k)=\int_{-\ell/2}^{\ell/2}e^{(k+\lambda/k)s}\left[\frac12 f'(s)+\frac\lambda kf(s)\right]ds.
$$
Divide the global relation by $E(-iak)$. Using $1+a+\bar a=0$ yields $E(-ik)/E(-iak)=e(\bar ak)$ and $E(-i\bar ak)/E(-iak)=e(-k)$. Hence
$$
\boxed{e(\bar ak)\Psi(k)+e(-k)\Psi(\bar ak)+\Psi(ak)=2iA(k),}
$$
where the requested known function is
$$
\boxed{A(k)=e(\bar ak)\Phi(k)+e(-k)\Phi(\bar ak)+\Phi(ak).}
$$
For an explicitly derivative-free form, put $H(k)=\int_{-\ell/2}^{\ell/2}e^{(k+\lambda/k)s}f(s)ds$ and $f_*=f(-\ell/2)=f(\ell/2)$. An <integration by parts> gives
$$
\Phi(k)=f_*\sinh\!\left(\frac\ell2(k+\lambda/k)\right)-\frac12\left(k-\frac\lambda k\right)H(k).
$$
The uniqueness qualification cannot be removed for every real $\lambda$: at a <Dirichlet Laplacian eigenvalue> a homogeneous addition need not preserve rotation invariance. The scalar reduced relation then applies to the rotation-averaged solution, but not necessarily to the originally chosen solution. Part (d) treats this genuine exceptional case.
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