Versal deformations of a Dirac type differential operator

If we are given a smooth differential operator in the variable x is an elem ent of R/2 pi Z, its normal form, as is well known, is the simplest form ob tainable by means of the Diff(S-1)-group action on the space of all such op erators. A versal deformation of this operator is a normal form for some pa rametric infinitesimal family including the operator. Our study is devoted to analysis of versal deformations of a Dirac type differential operator us ing the theory of induced Diff(S-1)-actions endowed with centrally extended Lie-Poisson brackets. After constructing a general expression for tranvers al deformations of a Dirac type differential operator, we interpret it via the Lie-algebraic theory of induced Diff(S-1)-actions on a special Poisson manifold and determine its generic moment mapping. Using a Marsden-Weinstei n reduction with respect to certain Casimir generated distributions, we des cribe a wide class of versally deformed Dirac type differential operators d epending on complex parameters.