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Based on lectures given at an advanced course on integrable systems at the Centre de Recerca Matemàtica in Barcelona, these lecture notes address three major aspects of integrable systems: obstructions to integrability from differential Galois theory; the description of singularities of integrable systems on the basis of their relation to bi-Hamiltonian systems; and the generalization of integrable systems to the non-Hamiltonian settings. All three sections were written by top experts in their respective fields.

Native to actual problem-solving challenges in mechanics, the topic of integrable systems is currently at the crossroads of several disciplines in pure and applied mathematics, and also has important interactions with physics. The study of integrable systems also actively employs methods from differential geometry. Moreover, it is extremely important in symplectic geometry and Hamiltonian dynamics, and has strong correlations with mathematical physics, Lie theory and algebraic geometry (including mirror symmetry). As such, the book will appeal to experts with a wide range of backgrounds.



Chapter 1. Integrable Systems and Difierential Galois Theory

At the end of the nineteenth century, Picard [25, 26], [27, Chapter XVII] and, in a clearer way, Vessiot in his PhD Thesis [30], created and developed a Galois theory for linear differential equations. This field of study, henceforth called Picard–Vessiot theory, was continued from the forties to the sixties of the twentieth century by Kolchin, through the introduction of the modern algebraic abstract terminology and the obtention of new important results, see [12] and references therein. Today, the standard reference of this theory is the monograph [29].
Juan J. Morales-Ruiz

Chapter 2. Singularities of bi-Hamiltonian Systems and Stability Analysis

The main topics in this section will be: Integrable Systems, the Lagrangian fibration and its singularities, the importance of singularities, and the non-degeneracy of singularities and their basic properties.
Alexey Bolsinov

Chapter 3. Geometry of Integrable non-Hamiltonian Systems

This text is an expanded version of the lecture notes for a minicourse taught by the author at the summer school “Advanced Course on Geometry and Dynamics of Integrable Systems” organized by Vladimir Matveev, Eva Miranda and Francisco Presas at Centre de Recerca Matemàtica (CRM) Barcelona, from September 9th to 14th, 2013. The aim of this minicourse was to present some geometrical aspects of integrable non-Hamiltonian systems. Here, the adjective non-Hamiltonian does not mean that the systems in question cannot be Hamiltonian, it simply means that we consider general dynamical systems which may or may not admit a Hamiltonian structure, and even when they are Hamiltonian we can sometimes forget about their Hamiltonian nature.
Nguyen Tien Zung
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