The Lie point symmetry generators admitted by systems of linear differential equations

Computing the Lie point symmetries ofsystems of linear differential equations can be prohibitively difficult. For homogeneous systems in Kovalevskaya form of order two or higher, this paper proves the existence of a basis of infinitesimal generators (as determined by Lie's algorithm) whose characteristic forms are homogeneous in the dependent variables of degree zero, one or two. Suppose Lie's algorithm yields a characteristic form of degree two; in this case, the system is second order. If it contains only ordinary differential equations (ODEs), its general solution is constructed from those of a givenfirst-order linear homogeneous system, of the same dimension, and a second-order linearscalar equation. Otherwise, coordinates are given in which its dimension (necessarily two or higher now) is lowered by one, leaving an inhomogeneous system of parametrized ODEs. Its homogeneous part is solved as in the previous case.