Let X be a hypergroup, K its compact subhypergroup and assume that (X, K) is a Gelfand
pair. Connections between finite dimensional varieties and K-polynomials on X are
discussed. It is shown that a K-variety on X is finite dimensional if and only if
it is spanned by finitely many K-monomials. Next, finite dimensional varieties on
affine groups over R-d, where d is a positive integer are discussed. A complete description
of those varieties using partial differential equations is given.