After giving examples of integrable matrix homogeneous ODE-systems we discuss non-abelian Laurent polynomials and an ODE-system of Kontsevich which we study for its integrability. Further studies towards the classification of such ODE systems and the investigation of Lax pairs of a given system lead to challenging bi-linear algebraic systems for undetermined coefficients to solve. To compute symmetries of order up to 13 for the Kontsevich system, very large linear algebraic systems had been solved. Techniques that have been used will be described.