Strict versions of integrable hierarchies in pseudodifference operators and the related Cauchy problems

In the algebra $Ps\Delta$ of pseudodifference operators, we consider two deformations of the Lie subalgebra spanned by positive powers of an invertible constant first-degree pseudodifference operator $\Lambda_0$. The first deformation is by the group in $Ps\Delta$ corresponding to the Lie subalgebra $Ps\Delta_{<0}$ of elements of negative degree, and the second is by the group corresponding to the Lie subalgebra $Ps\Delta_{\le0}$ 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 $\Lambda_0$ that respectively complements the Lie subalgebra $Ps\Delta_{<0}$ or $Ps\Delta_{\le0}$. This yields two integrable hierarchies associated with $\Lambda_0$, where the hierarchy of the wider deformation is called the strict version of the first because of the form of the Lax equations. For $\Lambda_0$ equal to the matrix of the shift operator, the hierarchy corresponding to the simplest deformation is called the discrete KP hierarchy. We show that the two hierarchies have an equivalent zero-curvature form and conclude by discussing the solvability of the related Cauchy problems.