Is it possible to solve the equation $x^a-a^y=xy$?

Question

How to solve the equation: $x^a-a^y=xy$ with the following conditions: $a\gt1,x\gt1,y\gt1$ and $x\in\mathbb{N}$,$y\in\mathbb{N}$,$a\in\mathbb{R}$? I found for $y(x,a)$ the following solution: $$y(x,a)=\frac{1}{x}x^{(a-1)}\ln(a)-\frac{1}{x}W\left(\frac{\ln(a)a^{x^(a-1)}}{x}\right)$$ where $W(z)$ is the LambertW function, but I don't know how to find integer solutions for $x$ and $y$

Answer 1

If you want an integer solution for $x,y$ I would be solving for $a$. For example, if $y\gt x\gt1$ are integers, the intermediate value theorem shows there is a solution with $a\in (1,2)$.

[Consider $f(a)=x^a-a^y$ and compute $f(1)=x-1\gt 0$ and $f(2)=x^2-2^y\lt 0$]