On the Generalized Maxwell–Bloch Equations

A new Hamiltonian structure of the Maxwell–Bloch equations is described. In this setting the Maxwell–Bloch equations appear as a member of a family of generalized Maxwell–Bloch systems. The family is parameterized by compact semi-simple Lie groups, the original Maxwell–Bloch system being the member corresponding to $SU(2)$. The Hamiltonian structure is then used in the construction of a new family of symmetries and the associated conserved quantities of the Maxwell–Bloch equations.