  We classify all special homogeneous curves. A special homogeneous curve
$\mathcal{H}$ consists of connected components of the hyperbolic points in the
level set $\{h=1\}$ of a homogeneous polynomial $h$ in two real variables of
degree at least three, and admits a transitive group action of a subgroup
$G\subset\mathrm{GL}(2)$ on $\mathcal{H}$ that acts via linear coordinate
change.
