A linear, almost periodic equation with an almost automorphic solution

We construct a scalar, first order, almost periodic 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 automor- phic but not almost periodic. Finally, we show that some equation in the hull of (*) admits an almost automorphic, nonalmost periodic solution.