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We consider the class of autonomous systems $\dot x=f(x)$, where $x \in {\bfR}^{2n}$, $f \in C^1({\bf R}^{2n})$ whose phase portrait is a Cartesian productof $n$ two-dimensional {\em centres}. We also consider perturbations of thissystem, namely $\dot x=f(x)+g(t,x)$, where $g \in C^1({\bf R}\times{\bfR}^{2n})$ 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 assumedto be $t^{-p}$ where $p>1$. We prove under this conditions the existence ofbounded solutions of the perturbed system and discuss their convergence tosolutions of the unperturbed system. This convergence depends on $p$. Moreover,we show that the original unperturbed system may be reduced to the form $\dotr=0$, $\dot\theta=A(r)$, and taking $r\in {\bf R}^m_{+}$, $\theta\in {\bfT}^n$, where ${\bf T}^n$ denotes the $n$-dimensional torus, we investigate themore general case of systems whose phase portrait is foliated on invarianttori. We notice that integrable Hamiltonian systems are of the same nature. Wegive also several examples, showing that the conditions of our theorems cannotbe improved

Asymptotic Integration of Nonlinear Systems of Differential Equations whose Phase Portrait is Foliated on Invariant Tori