  We consider the class of autonomous systems $\dot x=f(x)$, where $x \in {\bf
R}^{2n}$, $f \in C^1({\bf R}^{2n})$ whose phase portrait is a Cartesian product
of $n$ two-dimensional {\em centres}. We also consider perturbations of this
system, namely $\dot x=f(x)+g(t,x)$, where $g \in C^1({\bf R}\times{\bf
R}^{2n})$ and $g$ is asymptotically small, that is $g\Rightarrow 0$ as $t\to
+\infty$ uniformly with respect to $x$. The rate of decrease of $g$ is assumed
to be $t^{-p}$ where $p>1$. We prove under this conditions the existence of
bounded solutions of the perturbed system and discuss their convergence to
solutions of the unperturbed system. This convergence depends on $p$. Moreover,
we show that the original unperturbed system may be reduced to the form $\dot
r=0$, $\dot\theta=A(r)$, and taking $r\in {\bf R}^m_{+}$, $\theta\in {\bf
T}^n$, where ${\bf T}^n$ denotes the $n$-dimensional torus, we investigate the
more general case of systems whose phase portrait is foliated on invariant
tori. We notice that integrable Hamiltonian systems are of the same nature. We
give also several examples, showing that the conditions of our theorems cannot
be improved.
