  In this article, we characterize the Beurling and Model subspaces of the
Hardy-Hilbert space $H^2(\mathbb{D})$ invariant under the composition operator
$C_{\phi_a}f=f\circ\phi_a$, where $\phi_a(z) = az + 1 - a$ for $a \in (0,1)$ is
an affine self-map of the open unit disk $\mathbb{D}$. These operators have
universal translates (in the sense of Rota) and have attracted attention
recently due to their connection with the Invariant Subspace Problem (ISP) and
the classical Ces\`aro operator.
