Solve[{y z == -z x - z z, x z == -z y - z z, 
        x y == -z z - z z, x^2 + y^2 + z^2 == 1, 
        x + y + z == 0}, {x, y, z}]

Solve[{y z == a x - u, x z == a y - u, 
        x y == a z - u, x^2 + y^2 + z^2 == 1, 
        x + y + z == 0}, {x, y, z, a, u}]

Sqrt[216]