neutrinos may come from the direct decay of δ$_{L}$ and from the secondary decay of charged leptons. If a large decay rate of δ ++ L → ℓ $^{+}$ℓ+ is needed for explaining the current experimental data, the associated processes δ + L → ℓ $^{+}$ν$_{ℓ}$ and δ 0 L → ν$_{ℓ}$ν$_{ℓ}$ will give extra contributions to the neutrino flux, which can be more important than that neutrinos from the secondary decay of charged leptons. In Fig. 7, we compare the contributions to the total neutrino-induced muon flux from the direct neutrinos and that from the secondary lepton decays in the SH-II case. The difference can be clearly seen in the large angle region φ ≃ 30 ◦ in which the direct neutrino contribution is roughly five times as many as that from secondary charged lepton decays. In Fig. 7, we also give the predicted muon flux for the cases SH-I ∼ SH-IV. The results show that the SH-III case with S$_{D}$ decaying into 4 τ, 2 τ 2 ν$_{τ}$ and 4 ν$_{τ}$ predicts much larger flux than the other cases. The reason is that the direct neutrinos in this case have higher energy than that in other cases. This case can be tested in the future experiments with improved sensitivity by a factor of a few. Note that the 2 τ or 4 τ final state cases are now strongly disfavored by the latest Fermi-LAT gamma-ray data [36] even in the DM decay scenario. The predictions for the cases LH-I ∼ LH-IV are found to be very similar. The high energy gamma-rays generated by DM particle may come from two type of contributions. The first one is that the DM particle directly produces high energy gamma- rays through processes VIB and FSR. The second contribution comes from the ICS process. In this case, the low energy interstellar photons obtain energy from the high energy positrons and electrons. For the first contribution, the expected spectrum of high energy gamma-rays is given by where J$_{ΔΩ}$ is defined in the same way as for neutrino in Eq. (28). For a general observable region, it is convenient to choose the galactic coordinate ( b, l ) where b and l stand for the galactic latitude and longitude. Then J$_{ΔΩ}$ can be written as the following form where the integration upper limit s$_{max}$ = $^{√}$r 2 MW − r 2 $_{⊙}$(1 − cos 2 b cos 2 l ) + r$_{⊙}$ cos b cos l and d Ω = cos bdbdl . 18 J$_{ΔΩ}$ = 1 ρ$_{⊙}$r$_{⊙}$ 1 ΔΩ ∫ ΔΩ d Ω$^{∫}$ s$_{max}$ 0 ρ$^{(}$ $^{√}$r 2 ⊙ − 2 sr$_{⊙}$ cos b cos l + s 2 ) ds , (34) C. Gamma-rays d Φ$_{γ}$ dE$_{γ}$ = ρ$_{⊙}$r$_{⊙}$ 4 πm$_{D}$ J$_{ΔΩ}$ ( ∑ k Γ$_{k}$ dn k γ dE$_{γ}$$^{)}$ , (33)