There is a curious Diphantine equation showing up in my research:

$$ \frac{1}{a^2-1}+\frac{1}{b^2-1}=\frac{1}{c^2-1}+\frac{1}{d^2-1}. $$

I am trying to find its integer solutions where $a$, $b$, $c$ and $d\ge2$. Of course there are trivial ones satisfying $\{a, b\} = \{c, d\}$, and there are nontrivial ones. My question is then: for which positive integer $m$ is there a nontrivial solution such that $m$ divides $a$, $b$, $c$ and $d$?

Result of some experiments: For $m=3$ there is an infinite family constructed from solutions to the Pell equation $2x^2-y^2=1$. Let $(x,y)$ be a solution such that both $x$ and $y$ are odd (which makes up half the solutions). Then

$$ a=3x, b=\frac{3(3x^2-1)}{2}, c=d=3y $$

is a solution with common divisor 3. Of course then for $m=1$ there are infinitely many solutions (and many more additional ones to that family). But so far I haven't found any nontrivial even solutions (i.e., for $m=2$).