Abstract
In this paper we discuss and prove various properties of the algebra of pseudodifferential operators related to integrable hierarchies in this algebra, in particular the KPhierarchy and its strict version. Some explain the form of the equations involved or giveinsight in why certain equations in these systems are combined, others lead to additionalproperties of these systems like a characterization of the eigenfunctions of the linearizations ofthe mentioned hierarchies, the description of elementary Darboux transformations of bothhierarchies and the search for expressions in Fredholm determinants for the constructedeigenfunctions and their duals.
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