fourier analysis – Integer solutions of two trigonometric equations

Problem. Are there natural numbers $a,b,c$ such that $ab$ divides $c^2-1$ and both equations
$$sum_{k=1}^bsum_{n=1}^a x_{k,n}e^{ipi n/a}=sqrt{c-1}$$
and
$$sum_{k=1}^bsum_{n=1}^a y_{k,n}e^{ipi n/a}=sqrt{c+1}$$
have solutions $x_{k,n}$, $y_{k,n}$ in the set ${-1,1}$?

Remark. This problem was motivated by some problem on decompositions of finite groups.