  In a quantum Fermi system the energy per particle calculated at the second
order beyond the mean-field approximation diverges if a zero-range interaction
is employed. We have previously analyzed this problem in symmetric nuclear
matter by using a simplified nuclear Skyrme interaction, and proposed a
strategy to treat such a divergence. In the present work, we extend the same
strategy to the case of the full nuclear Skyrme interaction. Moreover we show
that, in spite of the strong divergence ($\sim$ $\Lambda^5$, where $\Lambda$ is
the momentum cutoff) related to the velocity-dependent terms of the
interaction, the adopted cutoff regularization can be always simultaneously
performed for both symmetric and nuclear matter with different
neutron-to-proton ratio. This paves the way to applications to finite nuclei.
