  Using bidifferential calculus, we derive a vectorial binary Darboux
transformation for the first member of the "negative" part of the AKNS
hierarchy. A reduction leads to the first "negative flow" of the NLS hierarchy,
which in turn is a reduction of a rather simple nonlinear complex PDE in two
dimensions, with a leading mixed third derivative. This PDE may be regarded as
describing geometric dynamics of a complex scalar field in one dimension, since
it is invariant under coordinate transformations in one of the two independent
variables. We exploit the correspondingly reduced vectorial binary Darboux
transformation to generate multi-soliton solutions of the PDE, also with
additional rational dependence on the independent variables, and on a plane
wave background. This includes rogue waves.
