GEOMETRICAL DESCRIPTION OF THE LOCAL INTEGRALS OF MOTION OF MAXWELL-BLOCH EQUATION

We represent a classical Maxwell-Bloch equation and relate it to positive part of the AKNS hierarchy in geometrical terms. The Maxwell-Bloch evolution is given by an infinitesimal action of a nilpotent subalgebra n+ of affine Lie algebra on a Maxwell–Bloch phase space treated as a homogeneous space of n+. A space of local integrals of motion is described using cohomology methods. We show that Hamiltonian flows associated with the Maxwell–Bloch local integrals of motion (i.e. positive AKNS flows) are identified with an infinitesimal action of an Abelian subalgebra of the nilpotent subalgebra n− on a Maxwell–Bloch phase space. Possibilities of quantization and lattice setting of Maxwell–Bloch equation are discussed.