  If $(G,K)$ is a Gelfand pair, with $G$ a Lie group of polynomial growth and
$K$ a compact subgroup of $G$, the Gelfand spectrum $\Sigma$ of the
bi-$K$-invariant algebra $L^1(K\backslash G/K)$ admits natural embeddings into
${\mathbb R}^n$ spaces as a closed subset.
  For any such embedding, define ${\mathcal S}(\Sigma)$ as the space of
restrictions to $\Sigma$ of Schwartz functions on ${\mathbb R}^n$. We call
Schwartz correspondence for $(G,K)$ the property that the spherical transform
is an isomorphism of ${\mathcal S}(K\backslash G/K)$ onto ${\mathcal
S}(\Sigma)$.
  In all the cases studied so far, Schwartz correspondence has been proved to
hold true. These include all pairs with $G=K\ltimes H$ and $K$ abelian and a
large number of pairs with $G=K\ltimes H$ and $H$ nilpotent.
  In this paper we study what is probably the simplest of the pairs with
$G=K\ltimes H$, $K$ non-abelian and $H$ non-nilpotent, with $H=M_2({\mathbb
C})$, the complex motion group, and $K=U_2$ acting on it by inner
automorphisms.
