A tricky math problem: restricting a in the equation a(b^2-1)=c^2+d^2 to force |b|=1?

This is a strange little question that is obscurely related to some research I’m trying to do right now. Basically, I have the equation a(b^2–1)=c^2+d^2. The question is, is there any value of a, or any restrictions I could put on a (i.e., it’s a perfect cube, etc.) that would force b to equal plus or minus 1. All variables can be assumed to be nonnegative integers. I was able to solve this in the case d=0 by specifying that a be a perfect square—that forces b = +/-1, c=0 as the only answer. I can’t tell if there’s any such recourse for me here.