We have,

x^2+y^2+z^2=10(x+y+z)

x^2+y^2+z^2=10x+10y+10z

x^2-10x+y^2-10y+z^2-10z=0

(x^2-10x+25)+(y^2-10y+25)+(z^2-10z+25)=75

(x-5)^2+(y-5)^2+(z-5)^2=75

a^2+b^2+c^2=75

By Fermat's 3 square theorem we can express it as a sum of three squares. We do that by trial an error (if you allow non-negatives),

0+25+36=75

Thus,

x=0,y=10,z=11

x=10,y=0,z=11

x=10,y=11,z=0

x=0,y=10,z=10

x=11,y=0,z=10

x=11,y=10,z=0