How can we get all positive integer solution for these two equations While solving a problem i was able to reduce the question to finding positive integer solutions that satisfy the equations:-
$x+y+z = 2020$ and $x^2 + y^2 + z^2 = xyz+4$.
I found 
$$(x,y,z)= (2,1009,1009),\ (1009,2,1009) \text{ and }(1009,1009,2)$$
as few solutions by guessing and hit and trial. How can I find other solutions (if they exist) to this equations

You can write second equation as $(x+y+z)^2-2(xy+yz+xz)=xyz+4$ that is $2020^2-4= xyz+ 2(xy+yz+xz)$. Now it could be useful to write RHS as a product. You can calculate $(2+x)(2+y)(2+z)=8+4(x+y+z)+ 2(xy+yz+xz)+xyz=8088+ 2(xy+yz+xz)+xyz$.
So we can write $(2+x)(2+y)(2+z)=8088+2020^2-4=2020^2+4\cdot 2020+4=2022^2$. Are you able to continue now?
