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 of the equations of motion. With respect to the case of anisochronous systems, there is the additional problem of finding the set of allowed rotation vectors, because they cannot be given a priori simply by looking at the unperturbed system. By considering the involved parameters (size of the perturbation, rotation vector and average action of a persisting invariant torus) as independent parameters we can introduce a function which is analytic in such parameters and only when the latter satisfy some constraint it becomes a solution: this can be regarded as a sort of singular implicit function problem. Therefore, although the dependence of the parameters, hence of the solution, upon the size of the perturbation is not smooth, in this way we construct explicitly the solution by using an absolutely convergent power series.