Application of fields formed by the Gauss periods to the investigation of cyclic diophantine equations

The question of the nonsolvability of the equation
$$ Z^*_l(x_0,x_1,\dots,x_t)=\prod^{l-1}_{i=0}\sum^t_{j=0}x_j\zeta^{ij}=Dl^wx^l,\quad (D\varphi(D)z,l)=1 $$
in rational integers $x_0,x_1,\dots,x_t,z$ satisfying certain additional conditions is investigated. Two cases are considered: 1) $l$ is a regular prime number and $0<t<l-1$; 2) $l$ is an irregular prime number, $l=fe+1$ ($f$ is prime), $l>c_0(f,t)$ and $l$ does not divide the Bernoulli numbers $B_{fk+1}$ ($k=1,3,\dots,e-1$), $B_{2fk}$ ($k=1,2,\dots,\frac{e}{2}-1$).