On some integrable generalizations of the continuous Toda system

In the present paper, we obtain some integrable generalizations of the continuous Toda system, generated by a flat connection form taking values in higher grading subspaces of the algebra of the area-preserving diffeomorphism of the torus $T^2$, and construct their general solutions. The grading condition which we use here, imposed on the connection, can be realized in terms of some holomorphic distributions on the corresponding homogeneous spaces.