Pairs of integer pairs with same lcm, gcd and mean The problem is to find all pairs of two distinct pairs(up to permutation) of integer(!) numbers $(a, b)$ and $(c, d)$ s.t. $$\operatorname{lcm}(a, b) = \operatorname{lcm}(c, d)$$
$$\gcd(a, b) = \gcd(c, d)$$ and 
$$\frac{a + b}{2} = \frac{c + d}{2}$$
It is easy to show that if both LCM and GCD are equal, then two pairs have the same product in absolute value and the same sum AND the same GCD.
There was a similar question on MathSE about natural numbers, answer to which is that such distinct pairs dont exist: Prove two distinct pairs of natural numbers with these properties do not exist
One such pair is ((-6; 35), (14;15)): their sums and LCMs and GCDs are all equal pairwise. How to find all such pairs?

We could first search for $\gcd(a,b)=1$ since after that, multiply all of (a,b,c,d) by f so that $\operatorname{lcm}(af,bf)=\operatorname{lcm}(cf,df)=\operatorname{lcm}(a,b).f^2,\gcd(af,bf)=\gcd(cf,df)=f, af+bf=cf+df$.
