Let $S$ be a set of all positive rational numbers $x$ such that $2x^2 - 1$ is a square, excluding $x=1$.

I am interested in computing as many as possible solutions in $S$ to either the following equations:

(1) $\qquad p^2 - 1 = (q^2 - 1)\cdot r^2$

(2) ...removed...

(3) $\qquad p^2 + q^2 = 1 + r^2$

What would a reasonable computational approach for finding solutions?

For the equation (1), I know one solution: $(p,q,r)=(\tfrac{373}{23}, \tfrac{85}{41}, \tfrac{205}{23})$ -- would it help to find more solutions?

EDIT: Equation (2) was not exactly the one I'm interested in. So I removed it.