Shatalov-Sternin's construction of complex WKB solutions and the associated Riemann surface

We re-examine Shatalov-Sternin's proof of existence of resurgent solutions of a linear ODE. In particular, we take a closer look at the "Riemann surface" (actually, a two-dimensional complex manifold) whose existence, endless continuability and other properties are claimed by those authors. We present a detailed argument for a part of the "Riemann surface" most relevant for the exact WKB method.