What are all the closed connected subgroups of $U(2)$? The trivial ones are $S^1$, $S^1\times S^1$ and $SU(2)$. Are there others?
Also, which ones have an invariant function that is linear? Clearly there is no such function for the subgroup $S^1\times S^1$. But for instance if the $S^1$ acts only on the first factor of $(z_1,z_2)$, then $z_2$ is an invariant linear function for the action.