From the first three equations, we conclude $x = \lambda /2$, $y = \lambda / 2$, and $z = \lambda / 2$. Hence, $x=y=z$. Now, from the bottom equation we get $$x=y=z=\frac{A}{3}.$$
The values of $x,y,z$ that minimize $x^2 + y^2 + z^2$ and satisfy $x+y+z=A$ are $x = y = z = A/3$.