How to deal with this system of equations? I am trying to investigate this system of equations:
$$
x^Tx-\mathbf{1}^T x=x^TP^tx=c
$$
where $c\in\mathbb{Z}^+$ is a given positive integer, $\mathbf{1}=\{1,1,\ldots,1\}$ is a vector with as much ones as elements in $x$ and nothing else, $t\in\mathbb{Z}^+$ is any positive integer except if $P^t=I$ and
$$P=\begin{pmatrix}
     0 & 1 & 0 & 0 & \cdots & 0 & 0 \\
     0 & 0 & 1 & 0 & \cdots & 0 & 0 \\
     0 & 0 & 0 & 1 & \cdots & 0 & 0 \\
     \vdots  & \vdots& \vdots & \ddots & \ddots & \vdots & \vdots \\
     0 & 0 & 0 & \cdots & 0 & 1 &0 \\
     0 & 0 & 0 & \cdots & 0 & 0 & 1 \\
     1 & 0 & 0 & \cdots & 0 & 0 & 0    
     \end{pmatrix}$$
is a permutation matrix that shifts the elements of $x$ by one position at a time.
Using alternative notation we can write the same equations as $\sum x_ix_{i+t}=c$ where $i+t$ should be done $\mod n$ (number of components in $x$). and we have different equation for $t=0$: $\sum x_i^2 - \sum x_i=c$.
Question
I am familiar with basics of linear algebra, matrices etc. but I don't think I am familiar with systems like this one. I need a bit of guidance to get a grasp, anything of the following might be of a great help:


*

*How do we generally deal with equations $x^TAx=c$? And with $x^TA^tx=c$?

*Is there a special name for this system of equations?

*Is there any theory around such systems, what keywords should I search for?

*More exactly - how can I analyze this system, what can I learn about $x$ from these equations?


Some results
By summing all of the equations one can find that
$$
  (\mathbf{1}^Tx)^2 - \mathbf{1}^Tx = n\lambda
$$
It's useful to assign the root to $c'$:
$$
  c'= \mathbf{1}^Tx = (1\pm\sqrt{1+4nc})/2
$$
It appears (numerical evidence) that:
$$\sum\limits_{\text{even }i} x_i = (c' \pm \sqrt{c'})/2$$
$$\sum\limits_{\text{odd }i} x_i = (c' \mp \sqrt{c'})/2$$
But I only managed to show that explicitly for $n=2$ and $n=4$. It is done by combining equations to eliminate anything but $x_1$ or $x_1+x_3$ respectively which than turns out to be described by a quadratic equation with the above roots.
Using similar approach I found that for $n=3$:
$$
  \frac{(c'- 2\sqrt{c'})}{3} \leq x_i \leq \frac{(c'+ 2\sqrt{c'})}{3}
$$
 A: A good area of mathematics to acquaint yourself with are those results dealing with the numerical range.  The numerical range of the matrix $P^t$ is 
$$\left\{\mathbf{y}^*P^t\mathbf{y} \mid \mathbf{y}\in\mathbb{C}^n,\ ||y||_2=1\right\},$$
which is a normalized version of the quadratic form that you're interested in.  Note that $P^t$ in your case is a normal matrix, which implies that the numerical range of $P^t$ is the convex hull of the eigenvalues of $P^t$ (and therefore $P$).  If a scalar $d\ne 0$ lies in the numerical range of $P^t$ for a particular unit vector $y,$ then you can construct a desired solution $x=\sqrt{\frac{c}{d}}y.$
A: Let confirm me if these equations are the correct representation of the constraints for the problem?. If you do we can move further into the closed solution.
In algebraic notation:
$$
\mathbf{P}_0: \sum_{i=1}^n x_i^2-x_i=c\\
\mathbf{P}_t,t:1...n-1: \sum_{i=1}^n x_ix_{i+j}=c\\
$$
Thus in matricial notation, we should have:
$$
\mathbf{P}_0: x^Tx-1^Tx=c\\
\mathbf{P}_t,t:1...n-1: x^TP_tx=c\\
$$
with the permutation matrix obtained from the identity or multiplying $t$ times:
$$
P_t=\left[I_{t...n}I_{1...n-1}\right]=P_1^t
$$
and as general case:
$$
\mathbf{P}_t,t:1...n-1: x^TP_tx-\delta_{0t}1^Tx=c
$$
The simplest $\mathbb{R}^2$ case, with $c=1$:
$$
2xy=1\\
x^2+y^2-x-y=1
$$
leads to :
$$
x=1/2 (2 \pm \sqrt 2)\\
y=1/2 (2 \mp \sqrt 2)
$$

Here it is clear that $\mathbf{P}_0$ is unit nD-sphere 2D-surface centered in $x_i^0=1/2$ and with radius $r=\sqrt{c-n/4}$:
$$
\mathbf{P}_0: (x-x^0)^T(x-x^0)=c-n/4\\
$$
And  $\mathbf{P}_t$ is nD-hyperboloid, 2D-surface, centered in the origin, and having $x_i=x_{i+t},i=1:n$ as the main axis.
