  We show that the $a$-parameterized family of the generalized
Constantin-Lax-Majda model, also known as the Okamoto-Sakajo-Wunsch model,
admits exact self-similar finite-time blowup solutions with interiorly smooth
profiles for all $a\leq 1$. Depending on the value of $a$, these self-similar
profiles are either smooth on the whole real line or compactly supported and
smooth in the interior of their closed supports. The existence of these
profiles is proved in a consistent way by considering the fixed-point problem
of an $a$-dependent nonlinear map, based on which detailed characterizations of
their regularity, monotonicity, and far-field decay rates are established. Our
work unifies existing results for some discrete values of $a$ and also explains
previous numerical observations for a wide range of $a$.
