  In this paper, we present a new derivation of the shape and evolution of the
integrated galaxy-wide initial mass function (IGIMF), incorporating explicitly
the effects of cosmic rays (CRs) as regulators of the chemical and thermal
state of the gas in the dense cores of molecular clouds. We predict the shape
of the IGIMF as a function of star formation rate (SFR) and CR density, and
show that it can be significantly different with respect to local estimates. In
particular, we focus on the physical conditions corresponding to IGIMF shapes
that are simultaneously shallower at high-mass end and steeper at the low-mass
end than a Kroupa IMF. These solutions can explain both the levels of
$\alpha$-enrichment and the excess of low-mass stars as a function of stellar
mass, observed for local spheroidal galaxies. As a preliminary test of our
scenario, we use idealized star formation histories to estimate the mean IMF
shape for galaxies of different $z=0$ stellar mass. We show that the fraction
of low-mass stars as a function of galaxy stellar mass predicted by these mean
IMFs agrees with the values derived from high-resolution spectroscopic surveys.
