The authors consider a \((n+1+m)\)-dimensional vector-filed \(N\) which, expressed in local coordinates \((I,y,\psi)\in \mathbb P=\mathbb I\times \mathbb Y\times \mathbb T^m\), (where \(\mathbb I\subset \mathbb R^n\), \(\mathbb Y\subset \mathbb R^n\) are open and connected; \(\mathbb T=\mathbb R/(2\pi \mathbb Z)\) is the standard torus), has the form \[N(I,y)=v(I,y)\partial_y+\omega(I,y)\partial_{\psi}.\] Such systems have been extensively investigated in the absence of the coordinate \(y\) and it is known that, after a small perturbing term is switched on, the normalized actions \(I\) turn to have exponential small variations compared to the size of the perturbation. The authors obtain the same result as for the classical situation. In addition, they observe that no trapping argument is needed, as no small denominator arises. They use the result to prove that the level sets of certain function called Euler integral have exponential small variations in a short time, closely to collisions.