A linear, almost periodic equation with an almost automorphic solution

We construct a scalar, first order, almost periodic ${\text{ODE}}( * )x + A(t)x = B(t)$ which admits bounded solutions, but no almost periodic solutions. Using this equation, we give an example of a two-dimensional, almost periodic system whose projective flow admits two minimal subsets, one of which is almost automorphic but not almost periodic. Finally, we show that some equation in the hull of $( * )$ admits an almost automorphic, nonalmost periodic solution.