Integral solutions of hyperboloid $x^2+y^2-z^2=1$ Are there integral solutions to the equation $x^2+y^2-z^2=1$?

Recall the Brahmagupta formula
$$(ad-bc)^2 + (ac+bd)^2 = (a^2+b^2)(c^2+d^2) = (ac-bd)^2 + (ad+bc)^2$$
Now find $a,b,c,d$ such that say $ad-bc = 1$. Then set $z = ac+bd$, $x=ac-bd$ and $y = ad+bc$ to get a solution.
For instance, one such one parameter family solution following the above procedure is
$$x = 2t, y = 2t^2-1, z = 2t^2$$
where $a = d = t, b = (t+1), c = (t-1)$ since $t^2 - (t^2-1) = 1$.
