The Generalized Mkdv Equations for B

We study a generalization of the hierarchy of mKdV equations (modi-ed KdV), which forms an integrable system. This generalization is based on a Lax operator associated to the equations, with principal components of degrees between ?3 and 0. The main result of the study is the com-mutation of the classical integrals of motion. For that purpose, one can construct an isomorphism between the space of jets of the system and a quotient of SL 2 (C((t))), so that the problem can be transposed over a geometric quotient. We can deene the integrals of motion by means of the monodromy matrices of the Lax operators and the description of Poisson brackets by the trigonometric r-matrix. The action of screening operators is on the densities and the intersection of the kernels is the set of integrals of motion by means of a diierential complex.