I tried applying the discrete Fourier transform to my system of equations and ended up with the following
$$\mathcal{F}[\mathbf b]_i = \mathcal{F}^*[\mathbf {x^*}]_i \mathcal{F}[\mathbf x]_i$$
Can this somehow be solved for x?
If it helps
In my case the $\mathbf b$ is a vector with real, positive integer values and takes form $(b+b',b,b,\ldots,b)$. As a result, $\mathcal{F}[\mathbf b]$ takes the same form and has positive integer entries as well.
Besides I am only interested in real non-negative integer solutions for $\mathbf x$.
An example
Just to make sure we are talking about the same thing here, this is what I mean by the discrete Fourier transform: $$\mathcal{F}[\mathbf b]_i = \sum_j b_j \exp\left\{-2\pi i\frac{i j}{n}\right\}$$
If $\mathbf b = (21,14,14)$ then $\mathcal{F}[\mathbf b] = (49,7,7)$. For that a $\mathbf x = (4,2,1)$ (actually any permutation of this) is a solution as $\mathcal{F}[\mathbf x] = (7,\frac{5-i\sqrt 3}{2},\frac{5+i\sqrt 3}{2})$. Could I have found $\mathbf x$ from that $\mathbf b$?