The triple x+y,x-y,2z is not necessarily primitive, so I think we should use (i) rather than the theorem in my first post?
<=> x+y,x-y,2z is a Pythagorean triple (but it's not necessarily primitive)
where d is natural number, m and n are integers of opposite parity with m>n>0 and gcd(m,n)=1.
Now how can we characterize ALL solutions of ? Is there any restriction on "d"?