  Let $G_0=K\ltimes\mathfrak p$ be the Cartan motion group associated with a
noncompact semisimple Riemannian symmetric pair $(G, K)$. Let $\frak a$ be a
maximal abelian subspace of $\mathfrak p$ and let $\p=\a+\q$ be the
corresponding orthogonal decomposition. A flat horocycle in $\p$ is a
$G_0$-translate of $\q$. A conical distribution on the space $\Xi_0$ of flat
horocycles is an eigendistribution of the algebra $\mathbb D(\Xi_0)$ of
$G_0$-invariant differential operators on $\Xi_0$ which is invariant under the
left action of the isotropy subgroup of $G_0$ fixing $\q$. We prove that the
space of conical distributions belonging to each generic eigenspace of $\mathbb
D(\Xi_0)$ is one-dimensional, and we classify the set of all conical
distributions on $\Xi_0$ when $G/K$ has rank one.
