  Using far (FUV) and near (NUV) ultraviolet photometry from guest investigator
programmes on the Galaxy Evolution Explorer (GALEX) satellite, optical
photometry from the MDM Observatory and optical integral-field spectroscopy
from SAURON, we explore the UV-linestrength relations of the 48 nearby
early-type galaxies in the SAURON sample. Identical apertures are used for all
quantities, avoiding aperture mismatch. We show that galaxies with purely old
stellar populations show well-defined correlations of the integrated FUV-V and
FUV-NUV colours with the integrated Mgb and Hbeta absorption linestrength
indices, strongest for FUV-NUV. Correlations with the NUV-V colour, Fe5015
index and stellar velocity dispersion are much weaker. These correlations put
stringent constraints on the origin of the UV-upturn phenomenon in early-type
galaxies, and highlight its dependence on age and metallicity. In particular,
despite recent debate, we recover the negative correlation between FUV-V colour
and Mg linestrength originally publicised by Burstein et al. (1988), which we
refer to as the "Burstein relation", suggesting a positive dependence of the
UV-upturn on metallicity. We argue that the scatter in the correlations is
real, and present mild evidence that a strong UV excess is preferentially
present in slow-rotating galaxies. We also demonstrate that most outliers in
the correlations are galaxies with current or recent star formation, some at
very low levels. We believe that this sensitivity to weak star formation,
afforded by the deep and varied data available for the SAURON sample, explains
why our results are occasionally at odds with other recent but shallower
surveys. This is supported by the analysis of a large, carefully-crafted sample
of more distant early-type galaxies from the Sloan Digital Sky Survey (SDSS),
more easily comparable with current and future large surveys.
