Additive generalizations of the Lax equation

We consider additive generalizations of matrix differential equations defined by particular representations of a Lie algebra. To the right-hand sides of such matrix differential equations we add appropriate terms chosen in such a way that it is possible to obtain important information about the dynamics without solving the equations. For example, we can write explicitly their first integrals and functions which are linear in time along solutions. Sometimes we can also predict the asymptotic behaviour of trajectories.