  [Abridged] We present a detailed study of the physical properties of the
molecular gas in a sample of 18 molecular gas-rich early-type galaxies (ETGs)
from the ATLAS$ 3D sample. Our goal is to better understand the star formation
processes occurring in those galaxies, starting here with the dense
star-forming gas. We use existing integrated $^{12}$CO(1-0, 2-1),
$^{13}$CO(1-0, 2-1), HCN(1-0) and HCO$^{+}$(1-0) observations and present new
$^{12}$CO(3-2) single-dish data. From these, we derive for the first time the
average kinetic temperature, H$_{2}$ volume density and column density of the
emitting gas, this using a non-LTE theoretical model. Since the CO lines trace
different physical conditions than of those the HCN and HCO$^{+}$ lines, the
two sets of lines are treated separately. We also compare for the first time
the predicted CO spectral line energy distributions (SLEDs) and gas properties
of our molecular gas-rich ETGs with those of a sample of nearby well-studied
disc galaxies. The gas excitation conditions in 13 of our 18 ETGs appear
analogous to those in the centre of the Milky Way. Such results have never been
obtained before for ETGs and open a new window to explore further
star-formation processes in the Universe. The conclusions drawn should
nevertheless be considered carefully, as they are based on a limited number of
observations and on a simple model. In the near future, with higher CO
transition observations, it should be possible to better identify the various
gas components present in ETGs, as well as more precisely determine their
associated physical conditions. To achieve these goals, we show here from our
theoretical study, that mid-J CO lines (such as the $^{12}$CO(6-5) line) are
particularly useful.
