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Singular complete integrability. (English) Zbl 0997.32024
This very interesting paper is concerned with the study of holomorphic vector fields in a neighbourhood of a singular point in \(\mathbb C^{n}\). Such a vector field is holomorphically normalizable if its linear part is not too wild and has enough formal first integrals as symmetries.
In order to prove this kind of result the author studies collections of vector fields which commute pairwise and are described by a Lie morphism from a complex commutative finite-dimensional Lie algebra to the algebra of vector fields in a neighbourhood of the singular point which is the origin of \(\mathbb C^{n}\). The result explains and generalizes some previous results e.g. of A. D. Bryuno [Trans. Mosc. Math. Soc. 25, 131-288 (1973); translation from Tr. Mosk. Mat. Obshch. 25, 119-226 (1971; Zbl 0263.34003) and ibid. 26, 199-239 (1974); translation from Tr. Mosk. Mat. Obshch. 26, 199-239 (1972; Zbl 0269.34003)] and J. Vey [Bull. Soc. Math. Fr. 107, 423-432 (1979; Zbl 0426.58022)]. An interesting geometric interpretation of the results is also given.

MSC:
32S65 Singularities of holomorphic vector fields and foliations
37F75 Dynamical aspects of holomorphic foliations and vector fields
17B66 Lie algebras of vector fields and related (super) algebras
17B80 Applications of Lie algebras and superalgebras to integrable systems
32G99 Deformations of analytic structures
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