Conservation and bifurcation of an invariant torus of a vector field

We consider small perturbations with respect to a small parameter $\varepsilon\ge0$ of a smooth vector field in $\mathbb R^{n+m}$ possessing an invariant torus $T_m$. The flow on the torus $T_m$ is assumed to be quasiperiodic with $m$ basic frequencies satisfying certain conditions of Diophantine type; the matrix $\Omega$ of the variational equation with respect to the invariant torus is assumed to be constant. We investigate the existence problem for invariant tori of different dimensions for the case in which $\Omega$ is a nonsingular matrix that can have purely imaginary eigenvalues.