Fourier-Laplace Transformation and Painlevé Systems

This paper investigates the relationship between different representations of Painlevé systems, specifically focusing on the Fourier-Laplace transformation. The core contribution is the description of this transformation between rank 3 and rank 2 D-module representations using formal microlocalization. This work is significant because it provides a deeper understanding of the structure of Painlevé systems, which are important in various areas of mathematics and physics. The conclusion about the existence of a biregular morphism between de Rham complex structures is a key result.