Let $Q$ be the ternary quadratic form $Q(x,y,z)=x^2+y^2-z^2$. For $k\in{\mathbb Z}$, denote by $E_k$ the equation $Q(x,y,z)=k$, to be solved in integers $x,y,z$.
It is easy to see that $E_k$ always has a solution (when $k$ is odd, $k=2t+1$, we have $k=Q(t+1,0,t)$, and when $k$ is even, $k=2t$, we have $k=Q(3t,4t-1,5t-1)$).
Is there a parametrization known giving all the solutions of $E_k$ ? Or parametrization with many variables, giving many solutions ?
The equation can be represented as a difference of squares.
$$x^2+y^2-z^2=k$$
Then asking a number you can write down the solution, presenting a $t$ as one of the multipliers.
$$y=y$$
$$x=\frac{y^2-k}{2t}-\frac{t}{2}$$
$$z=\frac{y^2-k}{2t}+\frac{t}{2}$$