  The question of global existence versus finite-time singularity formation is
considered for the generalized Constantin-Lax-Majda equation with dissipation
$-\Lambda^\sigma$, where $\widehat {{\Lambda}^\sigma}=|k|^\sigma$, both for the
problem on the circle $x \in [-\pi,\pi]$ and the real line. In the periodic
geometry, two complementary approaches are used to prove global-in-time
existence of solutions for $\sigma \geq 1$ and all real values of an advection
parameter $a$ when the data is small. We also derive new analytical solutions
in both geometries when $a=0$, and on the real line when $a=1/2$, for various
values of $ \sigma$. These solutions exhibit self-similar finite-time
singularity formation, and the similarity exponents and conditions for
singularity formation are fully characterized. We revisit an analytical
solution on the real line due to Schochet for $a=0$ and $\sigma=2$, and
reinterpret it terms of self-similar finite-time collapse. The analytical
solutions on the real line allow finite-time singularity formation for
arbitrarily small data, even for values of $\sigma$ that are greater than or
equal to one, thereby illustrating a critical difference between the problems
on the real line and the circle. The analysis is complemented by accurate
numerical simulations, which are able to track the formation and motion
singularities in the complex plane. The computations validate and extend the
analytical theory.
