A factorization for ℤ × ℤ-matrices yielding solutions of Toda-type hierarchies

In this paper one considers the problem of finding solutions to a number of Toda-type hierarchies. All of them are associated with a commutative subalgebra of the k×k-matrices. The first one is formulated in terms of upper triangular ℤ×ℤ-matrices, the second one in terms of lower triangular ones and the third is a combination of the two foregoing types. It is shown that in an appropriate group setting solutions of the linearization of these Lax equations can be constructed by using a Birkhoff-type decomposition in the relevant group