  Recently it was noted that different IceCube datasets are not consistent with
the same power law spectrum of the cosmic neutrinos: this is the IceCube
spectral anomaly, that suggests that they observe a multicomponent spectrum. In
this work, the main possibilities to enhance the description in terms of a
single extragalactic neutrino component are examined. The hypothesis of a
sizable contribution of Galactic high-energy neutrino events distributed as
$E^{-2.7}$ [ApJ 826, 185 (2016)] is critically analyzed and its natural
generalization is considered. The stability of the expectations is studied by
introducing free parameters, motivated by theoretical considerations and
observational facts. The upgraded model here examined has 1)~a Galactic
component with different normalization and shape $E^{- 2.4}$; 2)~an
extragalactic neutrino spectrum based on new data; 3)~a non-zero prompt
component of atmospheric neutrinos. The two key predictions of the model
concern the `high-energy starting events' collected from the Southern sky. The
Galactic component produces a softer spectrum and a testable angular
anisotropy. A second, radically different class of models, where the second
component is instead isotropic, plausibly extragalactic and with a relatively
soft spectrum, is disfavored instead by existing observations of muon neutrinos
from the Northern sky and below few 100 TeV.
