Does this system of equations have a solution?

Question

Over $\mathbb{C}$, with $\alpha$ being a root of $1+x+x^2$, does this system of equations have any exact solutions when $\lambda_4\neq 0$? \begin{align} -\lambda_1^2+\lambda_2\lambda_6&= \alpha^2\lambda_4, &\label{lambda4}\\ -\lambda_2\lambda_5+\lambda_1\lambda_6&=\lambda_3, &\label{lambda3}\\ \lambda_5&=-\alpha\lambda_4, & \label{lambda5}\\ \lambda_4\lambda_5-\lambda_3\lambda_6&= \lambda_2, &\\ \lambda_4\lambda_6-\lambda_1\lambda_3&=1, &\\ -\lambda_6^2+\lambda_1\lambda_5&=\lambda_1\lambda_4-\lambda_2\lambda_3. & \end{align} There are precisely $7$ solutions when $\lambda_4 = 0$.

Answer 1

Yes, it has many solutions for other $\lambda_4\neq 0$, for example \begin{align} \lambda_1 & = -1,\cr \lambda_2 & = -\frac{1+\sqrt{-3}}{2},\cr \lambda_3 & = -\alpha,\cr \lambda_4 & = 1,\cr \lambda_5 & = -\alpha,\cr \lambda_6 & = -\lambda_2. \end{align}