I have this system of $n$ non-linear equations in $n$ unknowns, arising out of my research problem. Given that $x_0=1$, I have to solve for $(x_1,x_2,\ldots,x_n).$
$$\sum_{i=0}^n x_i^2+2\sum_{j=1}^n\sum_{i=0}^{n-j}x_ix_{i+j}=1$$
$$\sum_{j=1}^n\sum_{i=0}^{n-j}~j^2~x_ix_{i+j}=0$$
$$\sum_{j=1}^n\sum_{i=0}^{n-j}~j^4~x_ix_{i+j}=0$$
$$\cdots ~\cdots ~\cdots$$
$$\sum_{j=1}^n\sum_{i=0}^{n-j}~j^{2(n-1)}~x_ix_{i+j}=0$$
Is there any way to find *EXACT* solutions to this system of non-linear equations?

**Reformulation of the problem**

I have reformulated the problem into a problem involving *matrix equations*, which is as follows:

Let $~\mathbf{y} = \left(\begin{array}{c} y_1 \\ y_2 \\ \vdots \\ y_n \end{array}\right)= \left(\begin{array}{cc} x_1 & x_2 & \cdots & x_{n-1} & x_n\\ x_2 & x_3 & \cdots & x_n & 0\\ \ldots & \ldots & \ldots & \ldots & \ldots\\ x_n & 0 & \cdots & 0 & 0 \end{array}\right) \left(\begin{array}{c} x_0 \\ x_1 \\ \vdots \\ x_{n-1} \end{array}\right)$, where $x_0=1$ and $x_n \neq 0$.

and $~\mathbf{M} = \left(\begin{array}{cc} 1^2 & 2^2 & \cdots & n^2\\ 1^4 & 2^4 & \cdots & n^4\\ \ldots & \ldots & \ldots & \ldots\\ 1^{2(n-1)} & 2^{2(n-1)} & \cdots & n^{2(n-1)} \end{array}\right)$

The system of equations is given by:

$$2\bigg(\sum_{i=1}^n y_i \bigg) +\bigg( \sum_{i=1}^n x_i^2 \bigg)=0$$ $$\mathbf{M}\mathbf{y}=\mathbf{0}$$

Thus we have a system of $1+(n-1)=n$ equations in $n$ unknowns (namely, $x_1,x_2,\ldots,x_n$). Note that we could have completely dispose of the $y_i$'s and use only $x_i$'s to write the equations, in which case they would look like beasts. Instead we choose to simplify.

The problem, as before, is to find *EXACT* solution for $(x_1,x_2,\ldots,x_n)$. There are algorithms available for approximate solutions. But for my research, I need only *EXACT* solution. Any help would be greatly appreciated.