  Strong gravitational lensing is regarded as the most precise technique to
measure the mass in the inner region of galaxies or galaxy clusters. In
particular, the mass within one Einstein radius can be determined with an
accuracy of order of a few percent or better, depending on the image
configuration. For other radii, however, degeneracies exist between galaxy
density profiles, precluding an accurate determination of the enclosed mass.
The source position transformation (SPT), which includes the well-known
mass-sheet transformation (MST) as a special case, describes this degeneracy of
the lensing observables in a more general way. In this paper we explore
properties of an SPT, removing the MST to leading order, i.e., we consider
degeneracies which have not been described before. The deflection field
$\boldsymbol{\hat{\alpha}}(\boldsymbol{\theta})$ resulting from an SPT is not
curl-free in general, and thus not a deflection that can be obtained from a
lensing mass distribution. Starting from a variational principle, we construct
lensing potentials that give rise to a deflection field
$\boldsymbol{\tilde{\alpha}}$, which differs from $\boldsymbol{\hat{\alpha}}$
by less than an observationally motivated upper limit. The corresponding mass
distributions from these 'valid' SPTs are studied: their radial profiles are
modified relative to the original mass distribution in a significant and
non-trivial way, and originally axi-symmetric mass distributions can obtain a
finite ellipticity. These results indicate a significant effect of the SPT on
quantitative analyses of lens systems. We show that the mass inside the
Einstein radius of the original mass distribution is conserved by the SPT;
hence, as is the case for the MST, the SPT does not affect the mass
determination at the Einstein radius. [...]
