  In order for a white dwarf (WD) to achieve the Chandrasekhar mass, M_C, and
explode as a Type Ia supernova (SNIa), it must interact with another star,
either accreting matter from or merging with it. The failure to identify the
types of binaries which produce SNeIa is the "progenitor problem". Its solution
is required if we are to utilize the full potential of SNeIa to elucidate basic
cosmological and physical principles. In single-degenerate models, a WD
accretes and burns matter at high rates. Nuclear-burning WDs (NBWDs) with mass
close to M_C are hot and luminous, potentially detectable as supersoft x-ray
sources (SSSs). In previous work we showed that > 90-99% of the required number
of progenitors do not appear as SSSs during most of the crucial phase of mass
increase. The obvious implication is that double-degenerate (DD) binaries form
the main class of progenitors. We show in this paper, however, that many
binaries that later become DDs must pass through a long-lived NBWD phase during
which they are potentially detectable as SSSs. The paucity of SSSs is therefore
not a strong argument in favor of DD models. Those NBWDs that are the
progenitors of DD binaries are likely to appear as symbiotic binaries for
intervals > 10^6 years. In fact, symbiotic pre-DDs should be common, whether or
not the WDs eventually produce SNeIa. The key to solving the progenitor problem
lies in understanding the appearance of NBWDs. Most do not appear as SSSs most
of the time. We therefore consider the evolution of NBWDs to address the
question of what their appearance may be and how we can hope to detect them.
