  We show that the De Gregorio model on the real line admits infinitely many
compactly supported, self-similar solutions that are distinct under rescaling
and will blow up in finite time. These self-similar solutions fall into two
classes: the basic class and the general class. The basic class consists of
countably infinite solutions that are eigenfunctions of a self-adjoint compact
operator. In particular, the leading eigenfunction coincides with the
finite-time singularity solution of the De Gregorio model recently obtained by
numerical approaches. The general class consists of more complicated solutions
that can be obtained by solving nonlinear eigenvalue problems associated with
the same compact operator.
