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We consider ($-\alpha$)-homogeneous solutions to the stationaryincompressible 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 theBeltrami and axisymmetric flows. The nonexistence result of the Beltrami($-\alpha$)-homogeneous solutions holds for all $\alpha<1$. We show thenonexistence of axisymmetric ($-\alpha$)-homogeneous solutions without swirlsfor $-2\leq \alpha<0$.  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 ofaxisymmetric Beltrami ($-\alpha$)-homogeneous solutions for $\alpha\in\mathbb{R}\backslash [0,2]$ and axisymmetric ($-\alpha$)-homogeneous solutionswith a nonconstant Bernoulli function for $\alpha\in \mathbb{R}\backslash[-2,2]$. This is the first existence result on ($-\alpha$)-homogeneoussolutions with no explicit forms.  For $2<\alpha<3$, constructed ($-\alpha$)-homogeneous solutions provide newexamples of the Beltrami/Euler flows in $\mathbb{R}^{3}\backslash\{0\}$ whoselevel sets of the proportionality factor/Bernoulli surfaces are nested surfacescreated by the rotation of the sign $``\infty"$.

Existence of homogeneous Euler flows of degree $-\alpha\notin [-2,0]$