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For all integers d f kf 1 and n suitably large we give explicit ex- amples of connected compact real-analytic submanifolds M HC n with the following properties: (1) Every non-trivial covering space of M is non-embeddable in the sense that it is not CR-isomorphic (with respect to its canonical CR-structure) to a CR- submanifold of CN for any N whatsoever. (2) M has fundamental group p1ðMÞ GZk 2 , where Z2 is the group of order two. (3) The covering spaces of M, indexed by the subgroups of p1ðMÞ, are pairwise CR-non-isomorphic. (4) M is a strongly pseudo- convex Cauchy-Riemann submanifold with CR-codimensionf d. (5) M is homoge- neous with respect to a compact linear subgroup G H GLðn;CÞ. (6) M is not locally a direct product of CR-manifolds of lower dimensions.

Non-embeddable CR-manifolds of higher codimension