  We consider analytic maps and vector fields defined in $\mathbb{R}^2 \times
\mathbb{T}^d$, having a $d$-dimensional invariant torus $\mathcal{T}$. The map
(resp. vector field) restricted to $\mathcal{T}$ defines a rotation of
frequency $\omega$, and its derivative restricted to transversal directions to
$\mathcal{T}$ does not diagonalize. In this context, we give conditions on the
coefficients of the nonlinear terms of the map (resp. vector field) under which
$\mathcal{T}$ possesses stable and unstable invariant manifolds, and we show
that such invariant manifolds are analyitic away from the invariant torus. We
also provide effective algorithms to compute approximations of
parameterizations of the invariant manifolds, and present some applications of
the results.
