If I have a quadratic form like $y^2 - x^2 - x = k$ none of the techniques I know work because of the nasty $x$. Note that homogenizing doesn't work because a solution of $Y^2 - X^2 - X Z = k Z^{(2)}$ does not lead to a solution of the original equation in integers, at least as far as I have been able to determine.
How does one understand the solution set of these equation? How can I solve them?