Let G G be a locally compact group, and let K K be a compact subgroup of Aut ( G ) {\operatorname {Aut}}(G) , the group of automorphisms of G G . There is a natural action of K K on the convolution algebra L 1 ( G ) {L^1}(G) , and we denote by L K 1 ( G ) L_K^1(G) the subalgebra of those elements in L 1 ( G ) {L^1}(G) that are invariant under this action. The pair ( K , G ) (K,G) is called a Gelfand pair if L K 1 ( G ) L_K^1(G) is commutative. In this paper we consider the case where G G is a connected, simply connected solvable Lie group and K ⊆ Aut ( G ) K \subseteq {\operatorname {Aut}}(G) is a compact, connected group. We characterize such Gelfand pairs ( K , G ) (K,G) , and determine a moduli space for the associated K K -spherical functions.