Solve the following nonlinear equations for $v$ and $w$

$Avv^TAw+Bvv^TBw=\lambda_1v+\lambda_2w$

$Aww^TAv+Bww^TBv=\lambda_1w+\lambda_2v$

$v^Tw=w^Tv=0$

$v^Tv=w^Tw=1$

where $\lambda_1, \lambda_2, \lambda_3$ are real. $A$, $B$ are n-by-n symmetric matrices. Furthermore, they are similar and each of them has exactly $n/2$ eigenvalues equal to $+1$ and $n/2$ eigenvalues equal to $-1$.

It is part of my attempt to minimize

$\sum_A |w^TAv|^2$

by Lagrange multiplier. Here A are all tensor products of Pauli matrices