  In this paper we study classification of homogeneous solutions to the
stationary Euler equation with locally finite energy. Written in the form $u =
\nabla^\perp \Psi$, $\Psi(r,\theta) = r^{\lambda} \psi(\theta)$, for $\lambda
>0$, we show that only trivial solutions exist in the range $0<\lambda<1/2$,
i.e. parallel shear and rotational flows. In other cases many new solutions are
exhibited that have hyperbolic, parabolic and elliptic structure of
streamlines. In particular, for $\lambda>9/2$ the number of different
non-trivial elliptic solutions is equal to the cardinality of the set
$(2,\sqrt{2\lambda}) \cap \mathbb{N}$. The case $\lambda = 2/3$ is relevant to
Onsager's conjecture. We underline the reasons why no anomalous dissipation of
energy occurs for such solutions despite their critical Besov regularity 1/3.
