Cauchy problems related to integrable matrix hierarchies

We discuss the solvability of two Cauchy problems in matrix pseudodifferential operators. The first is associated with a set of matrix pseudodifferential operators of negative order, a prominent example being the set of strict integral operator parts of products of a solution (L, {U_α } of the h [ð]-hierarchy, where h is a maximal commutative subalgebra of gl_n (C) . We show that it can be solved in the case of compatibility completeness of the adopted setting. The second Cauchy problem is slightly more general and relates to a set of matrix pseudodifferential operators of order zero or less. The key example here is the collection of integral operator parts of the different products of a solution {V_α } of the strict h [ð]-hierarchy. This system is solvable if two properties hold: the Cauchy solvability in dimension n and the compatibility completeness. Both conditions are shown to hold in the formal power series setting.