  We present Hubble Space Telescope (HST) and ground-based optical and
near-infrared observations of SN 2005hk and SN 2008A, typical members of the
Type Iax class of supernovae (SNe). Here we focus on late-time observations,
where these objects deviate most dramatically from all other SN types. Instead
of the dominant nebular emission lines that are observed in other SNe at late
phases, spectra of SNe 2005hk and 2008A show lines of Fe II, Ca II, and Fe I
more than a year past maximum light, along with narrow [Fe II] and [Ca II]
emission. We use spectral features to constrain the temperature and density of
the ejecta, and find high densities at late times, with n_e >~ 10^9 cm^-3. Such
high densities should yield enhanced cooling of the ejecta, making these
objects good candidates to observe the expected "infrared catastrophe," a
generic feature of SN Ia models. However, our HST photometry of SN 2008A does
not match the predictions of an infrared catastrophe. Moreover, our HST
observations rule out a "complete deflagration" that fully disrupts the white
dwarf for these peculiar SNe, showing no evidence for unburned material at late
times. Deflagration explosion models that leave behind a bound remnant can
match some of the observed properties of SNe Iax, but no published model is
consistent with all of our observations of SNe 2005hk and 2008A.
