  We study the effect of the splitting of neutron and proton effective masses
with isospin asymmetry on the properties of the Skyrme energy density
functional. We discuss the ability of the latter to predict observable of
infinite matter and finite nuclei, paying particular attention to controlling
the agreement with ab-initio predictions of the spin-isospin content of the
nuclear equation of state, as well as diagnosing the onset of finite size
instabilities, which we find to be of critical importance. We show that these
various constraints cannot be simultaneously fulfilled by the standard Skyrme
force, calling at least for an extension of its P-wave part.
