Let \(N\) be a connected and simply connected nilpotent Lie group, and let \(K\) be a subgroup of the automorphism group of \(N\). One says that the pair \((K, N)\) is a nilpotent Gelfand pair if \(L^1_K(N)\) is an abelian algebra under convolution. Such pairs arise naturally in harmonic analysis and representation theory of Lie groups, In this document the authors are interested in a class of Gelfand pairs which arise in analysis on nilpotent Lie groups. They establish a geometric model for the Gelfand spectra of nilpotent Gelfand pairs \((K,N)\) where the \(K\)-orbits in the center of \(N\) have a one-parameter cross section and satisfy a certain non-degeneracy condition. More specially, one shows that the one-to-one correspondence between the set \(\Delta(K,N)\) of bounded \(K-\)spherical functions on \(N\) and the set \({\mathcal A}(K,N)\) of \(K-\)orbits in the dual \(\mathfrak n^*\) of the Lie algebra for \(N\) is a homeomorphism for this class of nilpotent Gelfand pairs. This result had previously been shown for \(N\) a free group and \(N\) a Heisenberg group, and was conjectured to hold for all nilpotent Gelfand pairs in [C. Benson and G. Ratcliff, Transform. Groups 13, No. 2, 243–281 (2008; Zbl 1180.22011)].