Introduction

In order to make the results approachable and transparent, this introduction is quite detailed. Unlike the main body of the monograph (Chaps. 2–18), it starts in Sect. 1.1 with the Painlevé III equations, and explains immediately and concretely the space Mⁱⁿⁱ of initial conditions. Although this is quite long, it is just a friendly introduction to essentially well known facts on Painlevé III. Section 1.2 gives, equally concretely, the space M^mon of monodromy data (at this point, without explaining where it comes from). Section 1.3 presents the main results on real solutions. No special knowledge is required to understand these statements.