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

We represent a classical Maxwell-Bloch equation and relate it to posit ive part of the AKNS hierarchy in geometrical terms. The Maxwell-Bloch evolution is given by an infinitesimal action of a nilpotent subalgeb ra n(+) of affine Lie algebra s ($) over cap l(2) on a Maxwell-Bloch p hase space treated as a homogeneous space of n(+). A space of local in tegrals 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 hows) are identified with an infinitesimal action of an Abelian subalgebra of the nilpotent subalgebra n_ on a M axwell-Bloch phase space. Possibilities of quantization and lattice se tting of Maxwell-Bloch equation are discussed.