Find all the values of a,b,c s.t that x divides $a^2-cb,c^2-ab,b^2-ac$ where x is an integer

Question

if $$x = \frac{a^2-cb}{(a+b+c)(a^2+b^2+c^2-ab-ac-bc)},\\ y = \frac{c^2-ab}{(a+b+c)(a^2+b^2+c^2-ab-ac-bc)},\\ z = \frac{b^2-ac}{(a+b+c)(a^2+b^2+c^2-ab-ac-bc)}.$$ And a,b,c are integers, do you know of any way i can pick a,b,c s.t x,y and z are integers (i am not looking for a specific example but rather all the values of a,b,c s.t x,y and z are integers)? Nothing i try seems to work so i would greatly appreciate your help. Thanks in advance