Solutions:

x=y=z=1

x=y=z=-2

xz+y=2 (a)

yz+x=2 (b)

yx+z=2 (c)

from (a) y = 2-xz

sub into (b) and (c) gives

z+2x-x2z = 2 which factorises to give z(1+x)=2 (x /=1) (d)

x+2z-z2x = 2 which factorises to give x(1+z)=2 (z /=1) (e)

Substitute x from (e) in (d) gives:

z(1 + 2/(1+z))=2

z2 + z – 2 = 0

z = 1 or -2

x = 1 or -2

y = 1 or -2