  Einstein's gravity minimally coupled to free, massive, classical fundamental
fields admits particle-like solutions. These are asymptotically flat,
everywhere non-singular configurations that realise Wheeler's concept of a
geon: a localised lump of self-gravitating energy whose existence is anchored
on the non-linearities of general relativity, trivialising in the flat
spacetime limit. In arXiv:1708.05674 the key properties for the existence of
these solutions (also referred to as stars or self-gravitating solitons) were
discussed - which include a harmonic time dependence in the matter field -, and
a comparative analysis of the stars arising in the Einstein-Klein-Gordon,
Einstein-Dirac and Einstein-Proca models was performed, for the particular case
of static, spherically symmetric spacetimes. In the present work we generalise
this analysis for spinning solutions. In particular, the spinning
Einstein-Dirac stars are reported here for the first time. Our analysis shows
that the high degree of universality observed in the spherical case remains
when angular momentum is allowed. Thus, as classical field theory solutions,
these self-gravitating solitons are rather insensitive to the fundamental
fermionic or bosonic nature of the corresponding field, displaying similar
features. We describe some physical properties and, in particular, we observe
that the angular momentum of the spinning stars satisfies the quantisation
condition $J=m N,$ for all models, where $N$ is the particle number and $m$ is
an integer for the bosonic fields and a half-integer for the Dirac field. The
way in which this quantisation condition arises, however, is more subtle for
the non-zero spin fields.
