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In this paper we study the Darboux transformations of planar vector fields ofSchr\"odinger type. Using the isogaloisian property of Darboux transformationwe prove the "invariance" of the objects of the "Darboux theory ofintegrability". In particular, we also show how the shape invariance propertyof the potential is important in order to preserve the structure of thetransformed vector field. Finally, as illustration of these results, someexamples of planar vector fields coming from supersymmetric quantum mechanicsare studied.

Darboux Integrals for Schr\"odinger Planar Vector Fields via Darboux Transformations