How to prove symmetry of the following Maxwell-Bloch equations?

Question

I have the following Maxwell-Bloch equations:

$\dot{E}=-\alpha_{1} E+ k_{1}P$

$\dot{P}=-\alpha_{2}P+ k_{2}ED$

$\dot{D}=-\alpha_{3}(D-\lambda) -k_{3}EP$

In this system $\alpha_{1,2,3}$ are coefficients for different laser classes and $k_{1,2,3}$ constants. $\lambda$ is a pumping parameter

How do you prove that the system has symmetry?

I assume that you have to prove that when (E, P, D) is a solution, then so is (−E, −P, D) but how do you prove this mathematically?

Thanks!

Answer 1

Assume that $(E,P,D)$ solves the system. Then, for example, $E'=-\alpha_{1} E+ k_{1}P$ hence $(-E)'=-\alpha_{1}(-E)+ k_{1}(-P)$, that is, $(-E,-P,D)$ also solves the first equation. It remains to show that $(-E,-P,D)$ solves the second and third equations.