Is there a general [integer] solution for the family of hyperbolas with equation $x^2-xy-y^2=k$? I'm looking for integer solutions to a family of hyperbolas, with basic equation
$$x^2-xy-y^2=k.$$
Just wondering if there are any general solutions, or solutions given some restrictions on $k$.

You may convert this to the following equation:
$$(2x-y)^2-\mathbf{5} y^2=4k.$$
Let $X=2x-y$ and $Y=y$, then the equation becomes
$$X^2-\mathbf{5}Y^2=4k.$$
This is a Pell type equation. Now you can use the machinery to solve Pellian equations to answer this.
