In this paper we study a classification of linear systems on Lie groups with respect the conjugacy of the corresponding flows. We also describe stability according to Lyapunov exponents.
Our purpose is to use a Darboux homogenous derivative to understand the harmonic maps with values in homogeneous space. We present a characterization of these harmonic maps from the geometry of homogeneous space. Furthermore, our work covers all type of invariant geometry in homogeneous space.