  The recent observation of the blazar TXS 0506+056 suggests the presence of a
hard power-law component in the extraterrestrial TeV-PeV neutrino flux, in
agreement with the IceCube analysis on the 8-year through-going muon neutrinos
from the Northern Sky. This is slightly in tension with the soft power-law
neutrino flux deduced by the IceCube 6-year High Energy Starting Events data. A
possible solution to such a puzzle is assuming a two-component neutrino flux.
In this paper, we focus on the case where, in addition to an astrophysical
power-law, the second component is a pure neutrino line produced by decaying
Dark Matter particles. We investigate how to realize a neutrinophilic decaying
Dark Matter in an extension of the Standard Model. The main features of the
model are: i) the requirement of a new symmetry like a global $U(1)$ charge;
ii) the Dirac nature of active neutrinos; iii) a low-reheating temperature of
the Universe of about 1 TeV. We perform a likelihood statistical analysis to
fit the IceCube data according to the present Fermi-LAT gamma-rays constraints.
