  In this paper we study the Darboux transformations of planar vector fields of
Schr\"odinger type. Using the isogaloisian property of Darboux transformation
we prove the "invariance" of the objects of the "Darboux theory of
integrability". In particular, we also show how the shape invariance property
of the potential is important in order to preserve the structure of the
transformed vector field. Finally, as illustration of these results, some
examples of planar vector fields coming from supersymmetric quantum mechanics
are studied.
