Reflectionless operators and automorphic Herglotz functions

I am interested in canonical systems and Dirac operators that are reflectionless on an open set. In this situation, the half line $m$ functions are holomorphic continuations of each other and may be combined into a single function. By passing to the universal cover of its domain, we then obtain a one-to-one correspondence of these operators with Herglotz functions that are automorphic with respect to the Fuchsian group of covering transformations. I investigate the properties of this formalism, with particular emphasis given to the measures that are automorphic in a corresponding sense. This will shed light on the reflectionless operators as a topological space, on their extreme points, and on how the heavily studied smaller space of finite gap operators sits inside the (much) larger space.