Given $A_1,...,A_5$ all rank-1 matrices in $\mathbb{R}^{3\times3}$, consider the following system of equations: $$ v_1^\top A_i v_1 + v_2^\top A_i v_2 = 0 \quad \forall i \in\{1, .., 5\}$$ Where $v_1$ and $v_2$ are unknown orthonormal vectors in $\mathbb{R}^3$: $\lVert v_1 \rVert = \lVert v_2 \rVert=1$ and $v_1^\top v_2=0$.
Generally, multivariate quadratic systems can be tackled with Gröbner bases or homotopy continuation methods, as mentioned, for example, in this post.
However, assuming that a finite number of solutions exist, I am curious if the known structure of the problem can be used to simplify finding the solutions.
Any help is greatly appreciated.
