I have a 2 $\times$ 2 matrix of the form
mat[a_, b_, c_, d_, e_, x_, y_] := {{a - b*Cos[2*x]*Cos[2*y], c*(Cos[x] + Cos[y])}, {c*(Cos[x] + Cos[y]), d - e*Cos[2*x]*Cos[2*y]}};
where a,b,c,d,e are parameters and x,y are coordinates. I would like to find parameters such that one of the eigenvalues of the matrix, preferably the lower one, becomes flat in $|x|+|y|<\pi/2$. I have plotted these eigenvalues using
Manipulate[Plot3D[Eigenvalues[mat[a, b, c, d, e, x, y]], {x, -Pi, Pi}, {y, -Pi, Pi}, PlotRange -> {-7, 7}, ColorFunction -> "TemperatureMap"], {a, -1, 2}, {b, -1, 5}, {c, -1, 1}, {d, -1, 1}, {e, -1, 1}]
and I can roughly estimate that at $a=1$, $b=0$, $c=-0.465$, $b=0.505$, $e=-0.285$ the lower eigenvalue is almost flat. How can I find the exact parameters such that the eigenvalue is constant, or at least consists of four patches of constant values, in $|x|+|y|<\pi/2$.