Additive generalizations of the Lax equation

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