  In this paper we study the existence and linear stability of almost periodic
solutions for a NLS equation on the circle with external parameters. Starting
from the seminal result of Bourgain (2005) on the quintic NLS, we propose a
novel approach allowing to prove in a unified framework the persistence of
finite and infinite dimensional invariant tori, which are the support of the
desired solutions. The persistence result is given through a rather abstract
"counter-term theorem" `a la Herman, directly in the original elliptic
variables without passing to action-angle ones. Our framework allows us to find
"many more" almost periodic solutions with respect to the existing literature
and consider also non-translation invariant PDEs.
