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

SIGMA 11 (2015), 092, 17 pages      arXiv:1503.09023      https://doi.org/10.3842/SIGMA.2015.092
Contribution to the Special Issue on Algebraic Methods in Dynamical Systems

### Differential Galois Theory and Lie Symmetries

David Blázquez-Sanz a, Juan J. Morales-Ruiz b and Jacques-Arthur Weil c
a) Universidad Nacional de Colombia, Colombia
c) Université de Limoges, France

Received March 31, 2015, in final form November 11, 2015; Published online November 20, 2015

Abstract
We study the interplay between the differential Galois group and the Lie algebra of infinitesimal symmetries of systems of linear differential equations. We show that some symmetries can be seen as solutions of a hierarchy of linear differential systems. We show that the existence of rational symmetries constrains the differential Galois group in the system in a way that depends of the Maclaurin series of the symmetry along the zero solution.

Key words: linear differential system; Picard-Vessiot theory; differential Galois theory; infinitesimal symmetries.

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