Lie symmetries and differential Galois groups of linear equations (Q2759105)

Given a linear ordinary differential equation (ODE), one can simplify it if its Lie algebra of infinitesimal symmetries is known. The authors present a new algorithm that computes the Lie symmetries of a linear ODE. It is known that the dimension of the Lie algebra of symmetries of a linear ODE of order \(n\) equals 8 for \(n=2\) and \(n+1\), \(n+2\) or \(n+4\) for \(n>2\). The structure of the Lie algebra is known in all cases. This result is reproved in the paper, and a complete classification of the equations for the cases \(n+2\) and \(n+4\) is given. These two cases lead to exceptional differential Galois groups, which are also computed. There is no direct connection between the Galois group and the Lie algebra of symmetries; moreover, any algebraic subgroup of \(GL(n)\) occurs as a differential Galois group for a linear ODE of order \(n+1\).