  We reexamine the possibility of reconstructing the initial fluxes of
supernova neutrinos emitted in a future core-collapse galactic supernova
explosion and detected in a Megaton-sized water Cherenkov detector. A novel key
element in our method is the inclusion, in addition to the total and the
average energies of each neutrino species, of a "pinching" parameter
characterizing the width of the distribution as a fit parameter. We uncover in
this case a continuous degeneracy in the reconstructed parameters of supernova
neutrino fluxes at the neutrinosphere. We analyze in detail the features of
this degeneracy and show how it occurs irrespective of the parametrization used
for the distribution function. Given that this degeneracy is real we briefly
comment on possible steps towards resolving it, which necessarily requires
going beyond the setting presented here.
