We analyze Darboux transformations in very general settings formultidimensional linear partial differential operators. We consider all knowntypes of Darboux transformations, and present a new type. We obtain a fullclassification of all operators that admit Wronskian type Darbouxtransformations of first order and a complete description of all possiblefirst-order Darboux transformations. We introduce a large class of invertibleDarboux transformations of higher order, which we call Darboux transformationsof continued Type I. This generalizes the class of Darboux transformations ofType I, which was previously introduced. There is also a modification of thistype of Darboux transformations, continued Wronskian type, which generalizeWronskian type Darboux transformations.
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