I have a sample $X=(X_1, ...,X_n)\sim N(\mu,\sigma^2)$ with $\sigma^2$ known. The hypotheses are $H_0: \mu=\mu_0, H_1:\mu \neq \mu_0$. I know that in such a case an UMP test does not exist and so that I should proceed using a LR test, in order to find the rejection rule.
My professor also told me that for a sample distribution that belongs to the Exponential Family in the case of simple vs bilateral hypotheses an UMP test does not exist.
Thus, my question is theoretic: why does an UMP test does not exist in such cases? Which are the conditions under which an UMP test does not exist?
EDIT: I have found an example in which, instead, although the alternative hypothesis is bilateral, the UMP test exists.
A sample $X\sim U(0,\theta)$. The hypotheses are $H_0:\theta=\theta_0, H_1:\theta\neq \theta_0$.