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In this paper we present an a-posteriori KAM theorem for the existence of an$(n-d)$-parameters family of $d$-dimensional isotropic invariant tori withDiophantine frequency vector $\omega\in \mathbb R^d$, of type $(\gamma,\tau)$,for $n$ degrees of freedom Hamiltonian systems with $(n-d)$ independent firstintegrals in involution. If the first integrals induce a Hamiltonian action ofthe $(n-d)$-dimensional torus, then we can produce $n$-dimensional Lagrangiantori with frequency vector of the form $(\omega,\omega_p)$, with$\omega_p\in\mathbb R^{n-d}$. In the light of the parameterization method, wedesign a (modified) quasi-Newton method for the invariance equation of theparameterization of the torus, whose proof of convergence from an initialapproximation, and under appropriate non-degeneracy conditions, is the objectof this paper. We present the results in the analytic category, so the initialtorus is real-analytic in a certain complex strip of size $\rho$, and thecorresponding error in the functional equation is $\varepsilon$. We heavily usegeometric properties and the so called automatic reducibility to deal directlywith the functional equation and get convergence if $\gamma^{-2}\rho^{-2\tau-1}\varepsilon$ is small enough, in contrast with most of KAMresults based on the parameterization method, that get convergence if$\gamma^{-4} \rho^{-4\tau}\varepsilon$ is small enough. The approach issuitable to perform computer assisted proofs.

A Modified Parameterization Method for Invariant Lagrangian Tori for Partially Integrable Hamiltonian Systems