### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 058, 15 pages      arXiv:1901.09951      https://doi.org/10.3842/SIGMA.2019.058
Contribution to the Special Issue on Algebraic Methods in Dynamical Systems

### Linear Differential Systems with Small Coefficients: Various Types of Solvability and their Verification

Moulay A. Barkatou a and Renat R. Gontsov bc
a) Laboratoire XLIM (CNRS UMR 72 52), Département Mathématiques-Informatique, Université de Limoges, Faculté des Sciences et Techniques, 123 avenue Albert Thomas, F-87060 LIMOGES Cedex, France
b) Institute for Information Transmission Problems RAS, Bolshoy Karetny per. 19, build. 1, Moscow 127051, Russia
c) Moscow Power Engineering Institute, Krasnokazarmennaya 14, Moscow 111250, Russia

Received January 30, 2019, in final form July 31, 2019; Published online August 09, 2019

Abstract
We study the problem of solvability of linear differential systems with small coefficients in the Liouvillian sense (or, by generalized quadratures). For a general system, this problem is equivalent to that of solvability of the Lie algebra of the differential Galois group of the system. However, dependence of this Lie algebra on the system coefficients remains unknown. We show that for the particular class of systems with non-resonant irregular singular points that have sufficiently small coefficient matrices, the problem is reduced to that of solvability of the explicit Lie algebra generated by the coefficient matrices. This extends the corresponding Ilyashenko-Khovanskii theorem obtained for linear differential systems with Fuchsian singular points. We also give some examples illustrating the practical verification of the presented criteria of solvability by using general procedures implemented in Maple.

Key words: linear differential system; non-resonant irregular singularity; formal exponents; solvability by generalized quadratures; triangularizability of a set of matrices.

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