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By Daniel Bertrand, Benjamin Enriquez, Claude Mitschi, Claude Sabbah, and Reinhard Schäfke

This specific quantity is devoted to the reminiscence of Andrey A. Bolibrukh. It includes expository articles dedicated to a few elements of Bolibrukh's paintings, via ten refereed study articles. themes disguise advanced linear and nonlinear differential equations and quantum teams: monodromy, Fuchsian linear platforms, Riemann-Hilbert challenge, differential Galois concept, differential algebraic teams, multisummability, isomonodromy, Painlevé equations, Schlesinger equations, integrable platforms, KZ equations, advanced mirrored image teams, and root platforms. A e-book of the eu Mathematical Society. dispensed in the Americas via the yankee Mathematical Society.

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Additional resources for Differential Equations and Quantum Groups: Andrey A. Bolibrukh Memorial Volume

Example text

The additional case is the Goryachev–Chaplygin case, where, only on the manifold W0 (that is, for 2 = 0), K = r(p2 + q 2 ) + pγ3 is a first integral. The fact that the three and half mentioned cases are the only ones in which the system has an algebraic additional integral has been proved by Husson [10]. With a meromorphic first integral, it has been proved by Ziglin [21] using techniques which were invented by him [20] and are close to the ones I want to discuss here, namely properties of the monodromy groups of the (linearized) differential system.

Hence the Hénon–Heiles system is not Liouville integrable (at least with a rational first integral). I like this example very much, because this is really a beautiful academic example. Firstly, I have given two completely different arguments for the seemingly different properties (K) and (L). Secondly, it shows that the Galois group is something very rich. Of course, it contains the monodromy group of the variational equation, and hence also its Zariski closure. But, in this simple example, the Riemann surface is simply connected so that there is no monodromy at all.

29 29 31 32 34 34 35 36 37 38 39 40 41 42 43 44 46 46 I remember Andrei sitting in my office in Strasbourg and telling me his admiration for Sofia Kowalevskaya. I am not sure whether this was because I am a woman, or because he knew I had written a long paper on the “Kowalevski top” [7] or simply 28 Michèle Audin because he was very enthusiastic over her brilliant and romantic personality. I would have been very happy to discuss the topic of the present paper with him. It is with deep sadness, thinking both of Sofia and Andrei, that I reproduce (in Figure 1), the very first page of the beautiful paper [13] on the rigid body.

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