Integral solutions of $x^2+y^2+1=z^2$ I am interested in integral solutions of $$x^2+y^2+1=z^2.$$ Is there a complete theory comparable to the one for $x^2+y^2=z^2?$

For the equation: $qX^2+Y^2=Z^2+j$
In the case where a square: $a=\sqrt{\frac{j}{q}}$
Using equation Pell: $p^2-(q+1)s^2=1$
Then the solution can be written:
$X=2s(s\pm{p})L\pm{ap^2}+2aps\pm{a(q+1)s^2}=bL+af$
$Y=(p^2\pm2ps+(1-q)s^2)L\pm{ap^2}+2aps\pm{a(q+1)s^2}=cL+af$
$Z=(p^2\pm2ps+(q+1)s^2)L\pm{ap^2}+2a(q+1)ps\pm{a(q+1)s^2}=fL+at$
$L$ - any integer number given by us.
The most interesting thing is that these numbers are solutions of equations:
$qb^2+c^2=f^2$
$t^2-(q+1)f^2=\pm{q}$
If we use the equation Pell: $p^2-(q+1)s^2=k$
And substituting the solutions in the upper formula, we have solutions of the following equations.
$qb^2+c^2=f^2$
where: $c-b=k$
$t^2-(q+1)f^2=\pm{qk^2}$
True, I use this formula in reverse order. Find solutions of Pell's equation is much more complicated than the simple equations like Pythagorean triples. So find them and then have solutions of Pell's equation. The most interesting thing is that the solution of Pell related to Pythagorean triples.
