  Despite intense scrutiny, the progenitor system(s) that gives rise to Type Ia
supernovae remains unknown. The favored theory invokes a carbon-oxygen white
dwarf accreting hydrogen-rich material from a close companion until a
thermonuclear runaway ensues that incinerates the white dwarf. However,
simulations resulting from this single-degenerate, binary channel demand the
presence of low-velocity H-alpha emission in spectra taken during the late
nebular phase, since a portion of the companion's envelope becomes entrained in
the ejecta. This hydrogen has never been detected, but has only rarely been
sought. Here we present results from a campaign to obtain deep, nebular-phase
spectroscopy of nearby Type Ia supernovae, and include multi-epoch observations
of two events: SN 2005am (slightly subluminous) and SN 2005cf (normally
bright). No H-alpha emission is detected in the spectra of either object. An
upper limit of 0.01 M_Sun of solar abundance material in the ejecta is
established from the models of Mattila et al. which, when coupled with the
mass-stripping simulations of Marietta et al. and Meng et al. effectively rules
out progenitor systems for these supernovae with secondaries close enough to
the white dwarf to be experiencing Roche lobe overflow at the time of
explosion. Alternative explanations for the absence of H-alpha emission, along
with suggestions for future investigations necessary to confidently exclude
them as possibilities, are critically evaluated.
