I'm doing some number theory research and I came across these two Diophantine equations (created under my own transformations):
$$y^3 = ax^3 + bx$$ (where $a$ and $b$ are parameters)
$$z^3 = x^2 + xy$$
Has anyone seen these before? I'm trying to find references or links for these but I'm having trouble. I don't need to solve both, just one. I would like to solve them over the integers and (eventually) prove that there are only a finite number of solutions.
Edit: $x,y,z$ must be $> 1$.