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              <p>Abstract We present a general framework for Kadomtsev–Petviashvili (KP) hierarchies beyond standard settings, relying on algebraic and geometric extensions of the classical theory including diffeological algebras, non-commutative structures, and generalized derivations. Our approach encompasses both formal and non-formal operator settings and leads to a general criterion for the integrability and well-posedness of KP-like equations.</p>
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