= Solution
The <formal adjoint> of $\mathcal L=\Delta-k^2$ is itself. With the bilinear convention used in <Green second identity>, take any complex-valued $v$ satisfying $\mathcal Lv=0$. The <product rule for divergence> gives the family
$$
\boxed{\partial_x(vu_x-uv_x)+\partial_y(vu_y-uv_y)=v\mathcal Lu-u\mathcal Lv=0.}
$$
Equivalently, the <differential one-form>
$$
\omega_v=(vu_x-uv_x)\,dy-(vu_y-uv_y)\,dx
$$
is a <closed differential form>. Arbitrary <linear combinations> of adjoint solutions give further members of this family. One can also write it as $\nabla\cdot[v^2\nabla(u/v)]=0$ wherever $v\ne0$, but the first formula has no division and remains valid at zeros of $v$. For a sesquilinear convention replace $v$ by $\overline v$; no such conjugation is built into the bilinear identity used below.
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