**Author:**
Michele V. Bartuccelli, Guido Gentile

**Title:** *
Lindstedt series for perturbations
of isochronous systems. I. General theory
*

**Abstract: **
We give a proof of the persistence of invariant tori
for analytic perturbations of isochronous systems by using
the Lindstedt series expansion for the solutions.
With respect to the case of anisochronous systems,
there is the additional problem to find the set of
allowed rotation vectors for the invariant tori,
which can not given a priori simply
by looking at the unperturbed system, and which leads
to a sort of singular implicit function problem.
Albeit the solutions are not analytic in the size
of the perturbation, an analytic expansion for the
solution can be envisaged and successfully used
in order to explicitly construct the solution
as an absolutely convergent power series.

**Keywords:**
KAM invariant tori, isochronuos systems, perturbation theory,
Lindstedt series

Michele Bartuccelli

Department of Mathematics and Statistics,

University of Surrey

Guildford, GU2 7XH

e-mail: m.bartuccelli@eim.surrey.ac.uk

Guido Gentile

Dipartimento di Matematica

Università di Roma Tre

Largo San Leonardo Murialdo 1, 00146 Roma, Italy

e-mail: gentile@matrm3.mat.uniroma3.it