Existence of additional first-tier and shifted-square nests

Determine whether additional first-tier two-parameter nests and second-tier shifted-square nests exist beyond those presented in Tables 1 and 2 that yield solutions to A^4 + a B^4 = C^4 + a D^4.

Background

The paper identifies three first-tier nests and five second-tier shifted-square nests, which together cover almost all tested a ≤ 1000.

The authors explicitly pose whether further nests of either type exist, which could extend coverage and illuminate new structural patterns in the solution space.