  In our previous work (Assier \& Shanin, QJMAM, 2019), we gave a new spectral
formulation in two complex variables associated with the problem of plane-wave
diffraction by a quarter-plane. In particular, we showed that the unknown
spectral function satisfies a condition of additive crossing about its branch
set. In this paper, we study a very similar class of spectral problem, and show
how the additive crossing can be exploited in order to express its solution in
terms of Lam\'e functions. The solutions obtained can be thought of as tailored
vertex Green's functions whose behaviours in the near-field are directly
related to the eigenvalues of the Laplace-Beltrami operator. This is important
since the correct near-field behaviour at the tip of the quarter-plane had so
far never been obtained via a multivariable complex analysis approach.
