I've encountered the following system of equations from a previous question I asked (How to find the maximum and minimum points constrained by a function.)
$$ \begin{cases} e^{x^2-y^2}(2x) &= \lambda(2x)\\ e^{x^2-y^2}(-2y) &= \lambda(2y)\\ \end{cases} $$
It turns out that if I solve for $\lambda$ I get: $e^{x^2-y^2}=-e^{x^2-y^2}$, which gets me no solutions in $\mathbb{R}$.
But, if I compare both the equations I get: $\frac{x}{-y}=\frac{x}{y}$, which does have solutions in $\mathbb{R}$.
Maybe I'm not seeing it, but how is it that?
Your solution for $\lambda$ is wrong when $x=y=0$