  We present a variational approach which shows that the wave functions
belonging to quantum systems in different potential landscapes, are pairwise
linked to each other through a generalized continuity equation. This equation
contains a source term proportional to the potential difference. In case the
potential landscapes are related by a linear symmetry transformation in a
finite domain of the embedding space, the derived continuity equation leads to
generalized currents which are divergence free within this spatial domain. In a
single spatial dimension these generalized currents are invariant. In contrast
to the standard continuity equation, originating from the abelian $U(1)$-phase
symmetry of the standard Lagrangian, the generalized continuity equations
derived here, are based on a non-abelian $SU(2)$-transformation of a
Super-Lagrangian. Our approach not only provides a rigorous theoretical
framework to study quantum mechanical systems in potential landscapes
possessing local symmetries, but it also reveals a general duality between
quantum states corresponding to different Schr\"{o}dinger problems.
