x-y-z=0, so x=y+z. Let z=-1, then x=y-1, so, if y=3, x=2. x-y-z=2-3-(-1)=-1+1=0. So the trio is (x,y,z)=(2,3,-1).
Now let x<0, then y+z<0, y<-z Let z=2 and y=-5, so x=-5+2=-3. (x,y,z)=(-3,-5,2).
Let y=-4, so x=z-4, z=x+4. Let x=7, then z=11. (x,y,z)=(7,-4,11).