  The spectrum of a Gelfand pair of the form (K lx N, K), where N is a
nilpotent group, can be embedded in a Euclidean space Rd . The identification
of the spherical transforms of K-invariant Schwartz functions on N with the
restrictions to the spectrum of Schwartz functions on Rd has been proved
already when N is a Heisenberg group and in the case where N = N3,2 is the free
two-step nilpotent Lie group with three generators, with K = SO3 [2, 3, 11].
  We prove that the same identification holds for all pairs in which the
K-orbits in the centre of N are spheres. In the appendix, we produce bases of
K-invariant polynomials on the Lie algebra n of N for all Gelfand pairs (K lx
N, K) in Vinberg's list [27, 30].
  (The references numbers refers to the bibliography at the end of the article)
