I know that characterizing the solutions to an equation in a finite field is generally difficult, but I was wondering if anyone had anything to say about the equation

(ab)^2 + a^2 + b^2 = 0 mod p

I started by writing the equation as (a^2 + 1)(b^2 + 1) = 1 mod p, and also (ab + 1)^2 + (a-b)^2 = 1 mod p, but that hasn't seemed to help much. I know how to determine the number of solutions, but what I am more interested in is the (possible) implied relation between a and b.

Thanks!