After step 4 in the previous post, we know that

A + B = 6 and A = +/- B^2

If A = B^2, then B^2 + B - 6 = 0, making B either -3 or 2.

If A = -B^2, then B^2 - B + 6 = 0, and this has no real solutions.

So there are only two solutions in addition to (1,1), and these are (9,-3) and (4,2).