I am finding this question a little confusing.
For starters, the latter half seems to be in 2-D. I don't believe there is a problem sheet question on this. How can the potential have only two fourier components, when they are w.r.t differet G? A real (no complex) potential has an even no. of fourier components, in pairs of +,- G.
How does one use the secular equation, either from the first part of the question, or derived again from the schrodinger equation via fourier analysis in 2-D? I know if I am doing this in 1-D, I have an unperturbed wavefunction with fourier components psi_k, psi_k+G since these are waves that may interfere. In 2-D should I have psi_veck, psi_veck +vecG1, psi_veck + vecG2? the smallest set of interfering waves?
Is the 2-D free electron model likely to appear again?