  Atomic spectral lines for different isotopes are shifted, revealing a change
in the properties of the nucleus. For spinless nuclei such isotope shifts for
two distinct transitions are expected to be linearly related, at least at
leading order in a change of the nuclear mass and charge distribution. Looking
for a breaking of linearity in so-called King plots was proposed as a novel
method to search for physics beyond the standard model. In the light of the
recent experimental progress in isotope shift spectroscopy, the sensitivity of
these searches will become limited by the determination of the isotope masses
and/or by nuclear effects which may induce nonlinearities at an observable
level. In this work, we propose two possible generalizations of the traditional
King plot that overcome these limitations by including additional isotope shift
measurements, thus significantly extending the new physics reach of King plots
in a purely spectroscopy-driven approach.
