  The spectrum of a Gelfand pair $(K\ltimes N, K)$, where $N$ is a nilpotent
group, can be embedded in a Euclidean space. We prove that in general, the
Schwartz functions on the spectrum are the Gelfand transforms of Schwartz
$K$-invariant functions on $N$. We also show the converse in the case of the
Gelfand pair $(SO(3)\ltimes N_{3,2}, SO(3))$, where $N_{3,2}$ is the free
two-step nilpotent Lie group with three generators. This extends recent results
for the Heisenberg group.
