Many integrable systems from classical mechanics admit a complexification, where phase space and time are complexified, and the geometry of the (complex) momentum map is the best possible complex analogue of the geometry that appears in the Liouville Theorem (Theorem 4.28). Namely, in many relevant examples the generic complexified fiber is an affine part of an Abelian variety (a compact algebraic torus, see Chapter 5) and the integrable vector fields are translation invariant, when restricted to any of these tori. Such integrable systems are the main topic of this book, and we will call them algebraic completely integrable systems, following the original definition of Adler and van Moerbeke (see ).
KeywordsVector Field Meromorphic Function Irreducible Component Abelian Variety Integral Curve
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