The background to the question: 

$$a^4+b^4=c^4+d^4 \tag 1 $$

Tito Piezas & others have recently given some parametric solutions on Math stack exchange & Math overflow. In math literature there are parametric solutions given in Dickson’s book (vol 2), which includes solutions by Euler & others. Solutions are shown by Zajta (In a AMS journal paper). For degree two we have general solution for $a^2+b^2=c^2.$ For degree three $a^3+b^3=c^3+d^3$ two general solution has been given. One by Noam Elkies & second by Ajai Choudhry. Even though this problem of the quartic equation (# 1 above) has been around since the time of Euler (for over 200) years a general solution has been evasive. Eight relevant Links are shown below:

https://people.math.harvard.edu/%7Eelkies/4cubes.html


https://mathoverflow.net/questions/455960


https://math.stackexchange.com/questions/4779869


https://math.stackexchange.com/questions/4255315


http://euler.free.fr/identities.htm


https://mathworld.wolfram.com/DiophantineEquation4thPowers


https://www.ams.org/journals/mcom/1983-41-164


http://www.math.uni.wroc.pl/~jwr

Remark: 

There are more than a couple of dozen parametric solutions available for equation (1). There is a possibility that one of them could be a general solution or maybe not. There are two options. First is, someone needs to write an algorithm to see if the (1420 different) numerical solutions (link as shown above & given at Jaroslaw Wrobelewski website -uni.wroc) are satisfied by one of the published parametric solution. The other option is that someone needs to give a general solution along with a proof.  Any response to the above will be appreciated.