I am interested in finding every rational solution of $x (y - z) y (z - x) z (x - y) = t^2$ (expressed in homogenous form, to show its symmetry).
Among other approaches I am pursuing, it is clear that letting $t = x y z u$ (or a similar transform that flips all of x, y, z to the other side, each to the power 1, such as $t = x (y - z) u$) results in an equation which is quadratic in each of, x, y, z. So, holding two of these fixed, say x, y, and with u also fixed, one can in general leapfrog from one solution (x, y, z) to another (x, y, z') where z, z' are the two roots of the quadratic in z.
But naturally the next question is how many disjoint "3D lattices" of solutions formed in this way are needed to include every rational solution. Since u is fixed for every solution in any given lattice, I would imagine there must be an infinite number of these lattices, but that one can hop between them via the transformations that flip x, y, z to the right hand side and back again, if that makes sense.