Solvability of the number theoretic functional equation ${\varphi }_8\left(n\right)=\frac{n}{d}$

By using the exact formula for the generalized Euler function ${\varphi }_8\left(n\right),$ for one of the formulas, this paper studies the solvability for the equation ${\varphi }_8\left(n\right)=\frac{n}{d},$ where $n$ is a positive integer and $d$ is a positive factor of $n>8.$ Use elementary methods and techniques to obtain all its positive integer solutions $(n,d),$ when $\alpha =0,$ the positive integer solution of the equation is $(n,d)=\left(9,9\right)\left(11,11\right).$ When $\alpha =1$ and $\alpha =2,$ the equation has no positive integer solution.