  A small group of the newly discovered superluminous supernovae show broad and
slowly evolving light curves. Here we present extensive observational data for
the slow-evolving superluminous supernova LSQ14an, which brings this group of
transients to four in total in the low redshift Universe (z$<$0.2; SN 2007bi,
PTF12dam, SN 2015bn). We particularly focus on the optical and near-infrared
evolution during the period from 50 days up to 400 days from peak, showing that
they are all fairly similar in their light curve and spectral evolution.
LSQ14an shows broad, blue-shifted [O III] $\lambda\lambda$4959, 5007 lines, as
well as a blue-shifted [O II] $\lambda\lambda$7320, 7330 and [Ca II]
$\lambda\lambda$7291, 7323. Furthermore, the sample of these four objects shows
common features. Semi-forbidden and forbidden emission lines appear
surprisingly early at 50-70 days and remain visible with almost no variation up
to 400 days. The spectra remain blue out to 400 days. There are small, but
discernible light curve fluctuations in all of them. The light curve of each
shows a faster decline than $^{56}$Co after 150 days and it further steepens
after 300 days. We also expand our analysis presenting X-ray limits for LSQ14an
and SN2015bn and discuss their diagnostic power in interpreting their common
features. These features are quite distinct from the faster evolving
superluminous supernovae and are not easily explained in terms of only a
variation in ejecta mass. While a central engine is still the most likely
luminosity source, it appears that the ejecta structure is complex, with
multiple emitting zones and at least some interaction between the expanding
ejecta and surrounding material.
