  In this paper we prove a KAM theorem in infinite dimension which treats the
case of multiple eigenvalues (or frequencies) of finite order. More precisely,
we consider a Hamiltonian normal form in infinite dimension:\begin{equation}
\nonumberh(\rho)=\omega(\rho).r + \frac{1}{2} \langle \zeta,A(\rho)\zeta
\rangle,\end{equation}where $ r \in \mathbb{R}^n $, $\zeta=((p\_s,q\_s)\_{s \in
\mathcal{L}})$ and $ \mathcal{L}$ is a subset of $\mathbb{Z}$. We assume that
the infinite matrix $A(\rho)$ satisfies $A(\rho)= D(\rho)+N(\rho)$, where
$D(\rho) =\operatorname{diag} \left\lbrace \lambda\_{i} (\rho) I\_2,\: 1\leq i
\leq m\right\rbrace$ and $N$ is a bloc diagonal matrix. We assume that the size
of each bloc of $N$ is the multiplicity of the corresponding eigenvalue in
$D$.In this context, if we start from a torus, then the solution of the
associated Hamiltonian system remains on that torus. Under certain conditions
emitted on the frequencies, we can affirm that the trajectory of the solution
fills the torus. In this context, the starting torus is an invariant torus.
Then, we perturb this integrable Hamiltonian and we want to prove that the
starting torus is a persistent torus. We show that, if the perturbation is
small and under certain conditions of non-resonance of the frequencies, then
the starting torus is a persistent torus.
