### Abstract

We consider matrix equations of the form *d* **W**/*dz* = [S, **W**], where **S** is a matrix function of *z* that is embedded in a given Lie algebra*L*‐i.e., it is a curve in *L*. If the initial condition on **W** is in *L*, then **W** describes a curve in *L*. If the Killing form is used as a metric on *L*, then the behavior of the system is a pure rotation about an axis that is a function of *z*. A set of scalar invariants of such a system is obtained. These invariants form a set of conservation laws that the system obeys regardless of the detailed behavior of **S**(*z*).

If **S** describes a curve in some *L* _{1}, which is a semisimple subalgebra of the algebra*L* in which the whole system is embedded, then we can split the initial condition into two parts, one of which is in *L* _{1} and generates a solution in *L* _{1}, the other in a subspace that is orthogonal to *L* _{1} and that generates a curve that remains in the subspace. We can, then, obtain conservation laws that apply separately to the two parts.

The results have application to quantum mechanics since the density matrix obeys this type of equation. They also have application to coupled mode theory if we use, instead of the vector the corresponding power density spectrum matrix or the like.

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