  We consider ($-\alpha$)-homogeneous solutions to the stationary
incompressible Euler equations in $\mathbb{R}^{3}\backslash\{0\}$ for
$\alpha\geq 0$ and in $\mathbb{R}^{3}$ for $\alpha<0$. Shvydkoy (2018)
demonstrated the nonexistence of ($-1$)-homogeneous solutions and
($-\alpha$)-homogeneous solutions in the range $0\leq \alpha\leq 2$ for the
Beltrami and axisymmetric flows. The nonexistence result of the Beltrami
($-\alpha$)-homogeneous solutions holds for all $\alpha<1$. We show the
nonexistence of axisymmetric ($-\alpha$)-homogeneous solutions without swirls
for $-2\leq \alpha<0$.
  The main result of this study is the existence of axisymmetric
($-\alpha$)-homogeneous solutions in the complementary range $\alpha\in
\mathbb{R}\backslash [0,2]$. More specifically, we show the existence of
axisymmetric Beltrami ($-\alpha$)-homogeneous solutions for $\alpha\in
\mathbb{R}\backslash [0,2]$ and axisymmetric ($-\alpha$)-homogeneous solutions
with a nonconstant Bernoulli function for $\alpha\in \mathbb{R}\backslash
[-2,2]$. This is the first existence result on ($-\alpha$)-homogeneous
solutions with no explicit forms.
  For $2<\alpha<3$, constructed ($-\alpha$)-homogeneous solutions provide new
examples of the Beltrami/Euler flows in $\mathbb{R}^{3}\backslash\{0\}$ whose
level sets of the proportionality factor/Bernoulli surfaces are nested surfaces
created by the rotation of the sign $``\infty"$.
