Suppose I have an annulus $U\subset \mathbb{C}$ and a single-valued holomorphic function $V:U\to \mathbb{C}$.

I would like to know if there are (tractable) conditions on $V$ that ensure that the second order linear ODE
$$
\partial_{zz}^2 f+ V f=0
$$
admits a *single-valued* holomorphic solution $f: U\to \mathbb{C}$ (here $z$ is the usual complex coordinate).

Such a question is easy to answer for the first order linear ODE $$ \partial_{z} f+ V f=0 $$ Indeed, in this case separation of variables tells us that a necessary and sufficient condition is that the residue of $V$ is an integer.

I should note that taking $V=\alpha z^{-2}$ and $U$ the annulus $\lbrace z: 0<|z|<1\rbrace$ allows one to give explicit necessary and sufficient conditions on $\alpha$.

Where are questions of this sort treated? It seems like a classical problem...