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