Each term is a square, so characterize the variables as follows.
u > v > 0, u and v coprime, and u and v have different parity:
x ← u2 - v2
y2 ← 2uv
z ← u2 + v2
Since u and v are coprime, the second equation implies u and v are an odd square and twice a square. Conversely, any u and v meeting these criteria produce a solution.
Acceptable values for u and v are: 1, 2, 4, 8, 9, 16, 18, 25, 32, 36, 49, 50, etc. Here are the first four examples, when u and v are drawn from 1, 2, 8, and 9.
32 + 24 = 52
632 + 44 = 652
772 + 64 = 852
172 + 124 = 1452
s > t > 0, s and t coprime, and s and t of different parity:
u ← s2 + t2
v ← 2st
x ← 4st(s2+t2)
y ← s2-t2
z ← s4 + 6s2t2 + t4
The first example, with s=2 and t=1, is 402 + 34 = 412.
s > t > 0, s and t coprime, and s and t of different parity:
x ← s4 - 6s2t2 + t4
y ← 4st(s2-t2)
z ← s2 + t2
The first example, with s=2 and t=1, is 72 + 242 = 54.
Set s = 3 and t = 2 to get 1192 + 1202 = 134.