Shatalov-Sternin's construction of complex WKB solutions and the choice of integration paths

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" relevant for the exact WKB method. The present text is the author's article arXiv:0907.2934 rewritten from a different perspective.