Using the idea in theory of equation, we know that if sum of two roots are given and product of two roots are given then we can form a quadratic equation whose roots are required
Here, x+y=2 and xy=1+z^2
let us form an equation t^2 – (x+y)xt + yx=0
after solving, you will get x =1+zi or 1-zi
Similarly, for y=1-zi or 1+zi
So,only possible real pair is (1,1)
PS: I don’t know latex,so I couldn’t write the equation, if some one any source for latex,please tell