TITLE: Is there a generalization of Floquet theory to elliptic functions?
QUESTION [5 upvotes]: Hi, 
Consider  a system of linear differential equations
$$
{d f \over dz} = A(z) f,
$$
where $A(z)$ is a matrix-function. If $z \in \mathbb{R}$ and the function is periodic $A(z) = A(z + T)$, Floquet theorem applies. 
I am curious to know if there exists a generalization of Floquet theorem to the case, where $z \in \mathbb{C}$
and $A(z)$ is a doubly-periodic elliptic  function of $z$.
Thanks,
Victor

REPLY [2 votes]: Another, far reaching, aspect of Floquet theory in differential equations
$$a_0(z)f^{(n)}(z)+\cdots+a_nf(z)=0$$
with holomorphic coefficients is Fuchs theory of monodromy, where the leading coefficient $a_0$ has a zeros at $z_0$. You cannot solve a Cauchy problem at $z_0$, but you can solve it in a pointed disk $D\setminus z_0$. When you follow a circle around $z_0$, the coefficients look periodic.