Geometric Aspects of the Painlev\'e Equations ${\rm PIII(D_6)}$ and ${\rm PIII(D_7)}$

Abstract

The Riemann-Hilbert approach for the equations ${\rm PIII(D_6)}$ and ${\rm PIII(D_7)}$ is studied in detail, involving moduli spaces for connections and monodromy data, Okamoto-Painlev\'e varieties, the Painlev\'e property, special solutions and explicit B\"acklund transformations.