In this paper we present a classification, up to equivariant isomorphism, of $C^*$-dynamical systems $(A,{\mathbb R},\alpha )$ arising as inductive limits of directed systems $\{ (A_n,{\mathbb R},\alpha_n),\varphi_{nm}\}$, where each $A_n$ is a finite direct sum of matrix algebras over the continuous functions on the unit circle, and the $\alpha_n$s are outer actions generated by rotation of the spectrum.

An invariant is presented which classifies, up to equivariant isomorphism, $C^*$-dynamical systems arising as limits from inductive systems of elementary $C^*$-algebras on which the Euclidean motion group acts by way of unitary representations that decompose into finite direct sums of irreducibles.