Taking into account the rotation of mass-accreting white dwarfs (WDs) whose masses exceed the Chandrasekhar mass, we extend our new single degenerate model for the progenitors of Type Ia supernovae (SNe Ia), accounting for two types of binary systems: those with a main-sequence companion and those with a red-giant (RG) companion. We present a mass distribution of WDs exploding as SNe Ia, where the WD mass ranges from 1.38 to 2.3 M ⊙. These progenitor models are assigned to various types of SNe Ia. A lower mass range of WDs (1.38 M ⊙ < M WD ≲ 1.5 M ⊙), which are supported by rigid rotation, corresponds to normal SNe Ia. A variety of spin-down time may lead to a variation of brightness. A higher mass range of WDs (M WD ≳ 1.5 M ⊙), which are supported by differential rotation, corresponds to brighter SNe Ia such as SN 1991T. In this case, a variety of the WD mass may lead to a variation of brightness. We also show the evolutionary states of the companion stars at SN Ia explosions and pose constraints on the unseen companions. In the WD+RG systems, in particular, most of the RG companions have evolved to helium/carbon-oxygen WDs in the spin-down phase before the SN Ia explosions. In such a case, we do not expect any prominent signature of the companion immediately before and after the explosion. We also compare our new models with the recent stringent constraints on the unseen progenitors of SNe Ia such as SN 2011fe.
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