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

You can use the isomorphism T between SO(2,1) and SL(2) which is heavily used in physics. What it boils down to is that for every unimodular 2x2 matrix there is a 3x3 matrix of hyperbolic motions that has 3 unit (in Lorentzian sense) vectors for its columns.
So in those columns you not only get 2 solutions to your problem but also a solution to 
$x^2+y^2-z^2=-1$
Here it is in Mathematica code:
T[{{a_, b_}, {c_, d_}}] := {
{(a^2 + b^2 - c^2 - d^2)/2, (a^2 - b^2 - c^2 + d^2)/2, -ab + cd},
{-ac - bd, bd - ac,ad + bc},
 {(a^2 + b^2 + c^2 + d^2)/2, (a^2 - b^2 + c^2 - d^2)/2, -ab - cd}
}
So when you take for instance modular matrix m={{1,4},{2,9}}, you get the following columns:
u={-34,-38,51}
v={31,34,-46}
w={14,17,-22}
The first solves the "-1" equation, the second and third solve yours.
(https://arxiv.org/pdf/1508.00920.pdf)
