I have already done this twice, a long and short way and these proofs are already in the Smithsonian Institute too.
d**2/dx**2 + d**2/dy**2 = d**2z/dz**2
So x = y/(2x + 4) and y = x/(2y + 4) (new maths which is a complex transform of a rotation and translation of axes about point (0,0))
So x = y+2 and z = y + 2 and z = y + 5
and the answer is
x = 1 and y = 1 and z = 2 if and only if y is complex and z is free of all insternans like the world is about to turn.
(an insternan has no complex conjugate)