Essentially, when is $\dfrac{x^2 +y^2+1}{xy}$ a positive integer?
I've tried many approaches such as considering the above equation as a quadratic in $x$ or $y$ but I haven't had much success. The only solutions I could find were $x=1$, $y=1$ and $z=3$.
A similar problem: I know that the only solutions to $\dfrac{x^2+y^2}{xy}$ to equal a positive integer is for $x$ and $y$ to both be one. This is pretty much considering when a number, not necessarily an integer, and it's inverse sum to make a positive whole number. However, the problem is a slight variation, it asks when a number $\frac{p}{q}$ summed with it's inverse $\frac{q}{p}$ plus $\frac{1}{pq}$ is a whole number.
Using this post, suggested in the comments: Let $x$ and $y$ be positive integers such that $xy \mid x^2+y^2+1$.
we have $z=3$.
$x^2 - 3y\cdot x + (y^2+1)=0$, so the discriminant is a perfect square: $D=5y^2-4=t^2$. This is a Pell-type equation with infinitely many solutions: e.g., $y=2, t=4$, or $y=5, t=11$.
https://en.wikipedia.org/wiki/Pell%27s_equation