During some free time I had, I was wondering how to find the integer solutions $(x,y,z)$ to this generalized equation: $$z^2=axy+bx+cy+d$$ I am specifically looking for ways that do no involve factoring. And $a,b,c,d$ are all non-zero integers. I have no idea if it is easier or harder that for solving in two variables.
Edit: I have done some research and have concluded that it is a two-sheeted hyperboloid. I don't know if this helps with solving my question.