Fourier exponents of an almost periodic solution of differential equation

Consider an almost periodic solution $u$ of some differential equation (autonomous, nonautonomous, ordinary or in partial derivatives). There are methods to show the existence and uniqueness of such solution if we have almost periodic functions on right-hand side and initial conditions of differential equation. I'm interested in what we may know about the number of the Fourier exponents of $u$.

Sometimes we can show the containment of modulus (=the least additive subgroup of $\mathbb{R}$ containing the Fourier exponents), i.e., for example, that $\operatorname{mod}(u) \subset \operatorname{mod}(F,f_1,f_2)$, where $F$ is the right-hand side and $f_1$, $f_2$ are initial conditions. In particular, if there is finite integral base for the Fourier exponents of $(F,f_1,f_2)$ then we have finite integral base for the Fourier exponents of $u$ (but we can't say that the number of the Fourier exponents of $u$ is finite).

Question. In what case (=in what non-trivial class of differential equations) can we say that the number of the Fourier exponents of $u$ is finite and there is some connection between the exponents of $u$ and the exponents of $(F,f_1,f_2)$?

I give an example of the trivial case, where we can find the solution directly. Let $\omega$ be an irrational number and consider the differential equation $$y''(t)+y(t)=\sin(\omega t), \ y(0)=0, \ y'(0)=0.$$ The solution is $y(t)=\frac{\sin(\omega t)-\omega \sin(t)}{1-\omega^2}$.