k-Dirac operator and the Cartan-Kähler theorem for weighted differential operators

The k-Dirac operator is a first order differential operator which is natural to a particular class of parabolic geometries which include the Lie contact structures. A natural task is to understand the set of local null solutions of the operator at a given point. We will show that this set has a very nice and simple structure, namely we will show that there is a submanifold passing through the point such that any section defined on the submanifold extends locally to a unique null solution of the operator. This result also indicates that these parabolic geometries are naturally associated to a certain constant coefficient operator which has been studied in Clifford analysis and this is the original motivation for this paper. In order to prove the claim about the set of initial conditions for the k-Dirac operator we will adapt some parts of the theory of exterior differential systems and the Cartan-Kähler theorem to the setting of differential operators which are natural to geometric structures that are equipped with a filtration of the tangent bundle.