Elliptic tori in FPU non-linear chains with a small number of nodes

We revisit an algorithm constructing elliptic tori, that was originally designed for applications to planetary hamiltonian systems. The scheme is adapted properly work with models of chains $N+1$ particles interacting via anharmonic potentials, thus covering also the case FPU chains. After having preliminarily settled Hamiltonian in a suitable way, we perform sequence canonical transformations removing undesired perturbative terms by iterative procedure. This done using Lie series approach, explicitly implemented programming code help software package, which especially computer algebra manipulations. In cases $N=4,\, 8$, successfully apply our new construction tori wide sets parameter ruling size perturbation, i.e., total energy system. Moreover, explore stability regions surrounding 1D tori. compare semi-analytical results those provided numerical explorations FPU-model dynamics, where latter ones are obtained techniques based on so called frequency analysis. find procedure works up values same order magnitude respect maximal ones, detected methods.