Systems of second-order linear ODE’s with constant coefficients and their symmetries II. The case of non-diagonal coefficient matrices

We complete the analysis of the symmetry algebra LL for systems of n   second-order linear ODEs with constant real coefficients, by studying the case of coefficient matrices having a non-diagonal Jordan canonical form. We also classify the Levi factor (maximal semisimple subalgebra) of LL, showing that it is completely determined by the Jordan form. A universal formula for the dimension of the symmetry algebra of such systems is given. As application, the case n = 5 is analyzed.

► Symmetries of systems of second order linear ODE’s with constants coefficients. ► Analysis of possible dimensions. ► Symmetry algebras for non-diagonal systems. ► Dimension formula for arbitrary n and coefficient matrices. ► Structure of Levi factors of symmetry algebras.