On the number of solutions of the equation $(x_1+\ldots+x_n)^m=ax_1\ldots x_n$ in a finite field

We consider the equation $(x_1+\ldots +x_n)^m=ax_1\ldots x_n$, where $a$ is a nonzero element of the finite field $\mathbf F_q$, $n\ge 2$, and $m$ is a positive integer. Explicit formulas for the number of solutions of this equation in $\mathbf F_q^n$ under the condition $d\in\{1,2,3,6\}$, where $d=\mathrm{gcd}(m-n,q-1)$, are found. Moreover, we obtain formulas for the number of solutions for arbitrary $d>2$ if there exists positive integer $l$ such that $d\mid(p^l+1)$, where $p$ is the characteristic of $\mathbf F_q$.