Poisson geometry of the Maxwell-Bloch top system and stability problem

Dynamics of Maxwell-Bloch top system, that includes Maxwell-Bloch and Lorenz-Hamilton equations as particular cases, is studied in the framework Poisson geometry. Constants of motion as well as the relation of solution to that of pendulum are presented. Equilibrium states are determined and their complete stability analysis are performed. Results are applied to an optimal control problem on the Lie group G4.