Nekhoroshev type stability results for Hamiltonian systems with an additional transversal component

We prove exponential stability theorems of Nekhoroshev type for motion in the neighbourhood of an elliptic fixed point in Hamiltonian systems having an additional transverse component of arbitrary dimension, under certain conditions on this transverse component. We consider both the cases (i) of a strongly constrained motion, and (ii) of a weak perturbation.