**For 5 months!** I have been struggling to solve the following equations analytically without numeric method (ie, Newton method):

> **Main equation:**
> 
> $$
 \biggl(M^2-\cfrac{\mathbf{x^{\text{T}}}M^2\mathbf{x}}{\mathbf{x^{\text{T}}}\mathbf{x}}E\biggr)\mathbf{x}=\mathbf{1}
$$
> 
> **Constraint equations:**
> 
$$
\begin{cases}
 \mathbf{x^{\text{T}}1}=0 \\
\\
\mathbf{x^{\text{T}}x}=u 
\end{cases} $$
> 
> where $\{M,E\}\in\mathbf{R}^{n \times n}$ and $\{\mathbf{1},\mathbf{x}\}\in\mathbf{R}^n$ are defined, then $M$ is an arbitrary symmetric matrix, $E$
> is an identical matrix,
> $\mathbf{1}$ is all one vector, $\mathbf{x}$ is a
> variable vector and $u\in\mathbf{R}$ is a scalar.
> Furthermore, as a knowledge, the below equation form is called [Rayleigh
quotient][1] $R(M^2,\mathbf{x})$:
> 
> $$R(M^2,\mathbf{x}):=\cfrac{\mathbf{x^{\text{T}}}M^2\mathbf{x}}{\mathbf{x^{\text{T}}}\mathbf{x}}$$


Now, we attempt to estimate the $\mathbf{x}$. Does the analytic solution or method exist? My ability is shortage but, I guess that this problem has a beautiful solution. 



  [1]: https://en.wikipedia.org/wiki/Rayleigh_quotient