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There are many interesting dynamical systems in which degenerate invarianttori appear. We give conditions under which these degenerate tori have stableand unstable invariant manifolds, with stable and unstable directions havingarbitrary finite dimension. The setting in which the dimension is larger thanone was not previously considered and is technically more involved because insuch case the invariant manifolds do not have, in general, polynomialapproximations. As an example, we apply our theorem to prove that there aremotions in the $(n+2)$-body problem in which the distances among the first $n$bodies remain bounded for all time, while the relative distances between thefirst $n$-bodies and the last two and the distances between the last bodiestend to infinity, when time goes to infinity. Moreover, we prove that the finalmotion of the first $n$ bodies corresponds to a KAM torus of the $n$-bodyproblem.

Invariant manifolds of degenerate tori and double parabolic orbits to infinity in the (n+2)-body problem