Let $a,b,c\in\mathbb C$, and define $f(x,y)=ae^{i(x+y)}-b(e^{ix}+e^{iy})+c$ for $x,y\in[-\pi,\pi]$.
For a "generic" triple $a,b,c$ the set $\{f(x,y)=0\}$ consists of two points, but occasionally the zero set is a curve.
Is there any simple explanation of this fact? Can one define precise conditions for the degenerate situation to occur?
Example of the degenerate situation: $a=c=1,b=-2$. In this case the zero set is $ x=t,y=-i\log \biggr(-{1+2e^{it}\over {2+e^{it}}}\biggl)$