Strict Versions of Integrable Hierarchies in Pseudodifference Operators and the Related Cauchy Problems

In the algebra PsΔ of pseudodifference operators, we consider two deformations of the Lie subalgebra spanned by positive powers of an invertible constant first-degree pseudodifference operator Λ0. The first deformation is by the group in PsΔ corresponding to the Lie subalgebra Ps<0 of elements of negative degree, and the second is by the group corresponding to the Lie subalgebra PsΔ≤0 of elements of degree zero or lower. We require that the evolution equations of both deformations be certain compatible Lax equations that are determined by choosing a Lie subalgebra depending on Λ0 that respectively complements the Lie subalgebra PsΔ