If d is fixed, then any solution (a,b,c) yields a boring infinite set of solutions (a*k, b*k, c*k^2). Here are non-boring solutions with d=17: a=1, b=2, d=17, c=1 a=2, b=13, d=17, c=41 a=38, b=43, d=17, c=569 a=314, b=863, d=17, c=182209 a=859, b=1186, d=17, c=385241 a=2297, b=12134, d=17, c=35732401 If both b and d are fixed, then a^4+b^4=d*c^2 is an "elliptic curve" enabling you to generate an infinite set of rational solutions (a,c) from just a few (indeed usually only one) starter solutions by using the elliptic curve group. This observation is much more powerful than Desbove's method of getting new solutions from old. Then by multiplying by appropriate denominators we get an infinite set of integer solutions now with b not fixed. I do not know of any parameterized family of non-boring solutions.