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

$Avv^TAw=\lambda_1v+\lambda_2w$

$Aww^TAv=\lambda_1w+\lambda_2v$

$v^Tw=w^Tv=0$

$v^Tv=w^Tw=1$

where $\lambda_1, \lambda_2, \lambda_3$ are real. $A$ is a symmetric matrix.

How would you generalize to the case

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

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

Where both A and B are symmetric? Would it help if they are also similar and each of them has exactly $n/2$ eigenvalues equal to $+1$ and $n/2$ eigenvalues equal to $-1$?