  The flow of fallback matter being shocked and repelled back by an energy
deposition from a central object is discussed by using newly found self-similar
solutions. We show that there exists a maximum mass accretion rate if the
adiabatic index of the flow is less than or equal to 4/3. Otherwise we can find
a solution with an arbitrarily large accretion rate by appropriately shrinking
the energy deposition region. Applying the self-similar solution to supernova
fallback, we discuss how the fate of newborn pulsars or magnetars depends on
the fallback accretion and their spin-down power. Combining the condition for
the fallback accretion to bury the surface magnetic field into the crust, we
argue that supernova fallback with a rate of dM_{fb}/dt ~ 10^{-(4-6)} M_sun /s
could be the main origin of the diversity of Galactic young neutron stars,
i.e., rotation-powered pulsars, magnetars, and central compact objects.
