There are infinitely many solutions.
Algebraically, we can eliminate [tex]z[/tex] and try to solve for [tex]x,y[/tex]:
[tex]x+y+z=1\implies z=1-x-y[/tex]
Then
[tex]x^2+y^2+(1-x-y)^2=1[/tex]
[tex]\implies x^2+y^2+1-2x+x^2-2y+y^2+2xy=1[/tex]
[tex]\implies x^2-x+y^2-y+xy=0[/tex]
which is the equation of an ellipse.
