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The spectrum of a Gelfand pair of the form (K lx N, K), where N is anilpotent group, can be embedded in a Euclidean space Rd . The identificationof the spherical transforms of K-invariant Schwartz functions on N with therestrictions to the spectrum of Schwartz functions on Rd has been provedalready when N is a Heisenberg group and in the case where N = N3,2 is the freetwo-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 theK-orbits in the centre of N are spheres. In the appendix, we produce bases ofK-invariant polynomials on the Lie algebra n of N for all Gelfand pairs (K lxN, K) in Vinberg's list [27, 30].  (The references numbers refers to the bibliography at the end of the article)

Nilpotent Gelfand Pairs and Spherical Transforms of Schwartz Functions I. Rank-One Actions on the Centre