  We present the results of a study of the stellar and accretion properties of
the (almost) complete sample of class II and transitional YSOs in the Lupus I,
II, III and IV clouds, based on spectroscopic data acquired with the
VLT/X-Shooter spectrograph. Our study combines the dataset from our previous
work with new observations of 55 additional objects. We have investigated 92
YSO candidates in total, 11 of which have been definitely identified with giant
stars unrelated to Lupus. The stellar and accretion properties of the 81 bona
fide YSOs, which represent more than 90% of the whole class~II and transition
disc YSO population in the aforementioned Lupus clouds, have been homogeneously
and self-consistently derived, allowing for an unbiased study of accretion and
its relationship with stellar parameters. The accretion luminosity, Lacc,
increases with the stellar luminosity, Lstar, with an overall slope of ~1.6,
similar but with a smaller scatter than in previous studies. There is a
significant lack of strong accretors below Lstar~0.1Lsun, where Lacc is always
lower than 0.01Lstar. We argue that the Lacc-Lstar slope is not due to
observational biases, but is a true property of the Lupus YSOs. The
logMacc-logMstar correlation shows a statistically significant evidence of a
break, with a steeper relation for Mstar<0.2Msun and a flatter slope for higher
masses. The bimodality of the Macc-Mstar relation is confirmed with four
different evolutionary models used to derive the stellar mass. The bimodal
behaviour of the observed relationship supports the importance of modelling
self-gravity in the early evolution of the more massive discs, but other
processes, such as photo evaporation and planet formation during the YSO's
lifetime, may also lead to disc dispersal on different timescales depending on
the stellar mass. We also refined the empirical Lacc vs. Lline relationships.
