UMP tests for one- and two-sided alternatives on $\sigma^2$

Let $x_1,x_2,\dots,x_n$ be a random sample from $N(\mu,\sigma^2)$ distribution with $\mu$ known. Consider the following testing problems for $N(\mu,\sigma^2)$:

I. $H_0:\sigma^2=\sigma_0^2$ against $H_1:\sigma^2<\sigma_0^2$

II. $H_0:\sigma^2=\sigma_0^2$ against $H_1:\sigma^2>\sigma_0^2$

III. $H_0:\sigma^2=\sigma_0^2$ against $H_1:\sigma^2\ne\sigma_0^2$

The UMP level $\alpha$ test exists for: