  The main obstacle for gravitational lensing to determine accurate masses of
deflectors, or to determine precise estimates for the Hubble constant, is the
degeneracy of lensing observables with respect to the mass-sheet transformation
(MST). The MST is a global modification of the mass distribution which leaves
all image positions, shapes and flux ratios invariant, but which changes the
time delay. Here we show that another global transformation of lensing mass
distributions exists which almost leaves image positions and flux ratios
invariant, and of which the MST is a special case. Whereas for axi-symmetric
lenses this source position transformation exactly reproduces all strong
lensing observables, it does so only approximately for more general lens
situations. We provide crude estimates for the accuracy with which the
transformed mass distribution can reproduce the same image positions as the
original lens model, and present an illustrative example of its performance.
This new invariance transformation most likely is the reason why the same
strong lensing information can be accounted for with rather different mass
models.
