We consider small perturbations with respect to a small parameter ε ≥ 0 of a smooth vector field in 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 Ω 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 Ω is a nonsingular matrix that can have purely imaginary eigenvalues.