I have a function in 2 variables $F(x,y)$, which also contains the constant parameters $\epsilon_{1}$, $\epsilon_{2}$,$\beta$. It is defined as:
F[x_,y_]:=(3 - 3 beta (e1 + e2) + 3/8 beta^2 (3 e1^2 + 2 e1 e2 + 3 e2^2) + (-1 + beta e1) (-4 + beta (e1 + 3 e2)) Cos[(3 x)/2] Cos[(Sqrt[3] y)/2] + 1/8 (-4 + beta (e1 + 3 e2))^2 Cos[Sqrt[3] y])
My goal is to find the solutions to the equation $F(x,y)=0$, i.e. the coordinates of the points (x,y) on the plane (as a function of the parameters $\epsilon_{1}$, $\epsilon_{2}$,$\beta$) where the function is 0. I'm looking for an analytic expression of these coordinates.
I've tried with
FindInstance[F[x,y] == 0, {x, y}, Reals]
or something similar.
The problem is that it either returns me an error because of the presence of the parameters $\epsilon_{1}$, $\epsilon_{2}$,$\beta$, or it gives me a solution for these parameters as well, treating them as variables just like x,y.
So how can I find the solutions?