  In this paper, we prove the existence of self-similar algebraic spiral
solutions for 2-D incompressible Euler equations for the initial vorticity of
the form $|y|^{-\frac1\mu}\ \mathring{\omega}(\theta)$ with $\mu>\frac12$ and
$\mathring{\omega}\in L^1(\mathbb T)$ satisfying $m$-fold symmetry ($m\geq 2$)
and a dominant condition. As an important application, we prove the existence
of weak solution when $\mathring{\omega}$ is a Radon measure on $\mathbb T$
with $m$-fold symmetry, which is related to the vortex sheet solution.
