  We revisit recent results on integrable cases for higher-dimensional
generalizations of the 2D pentagram map: short-diagonal, dented, deep-dented,
and corrugated versions, and define a universal class of pentagram maps, which
are proved to possess projective duality. We show that in many cases the
pentagram map cannot be included into integrable flows as a time-one map, and
discuss how the corresponding notion of discrete integrability can be extended
to include jumps between invariant tori. We also present a numerical evidence
that certain generalizations of the integrable 2D pentagram map are
non-integrable and present a conjecture for a necessary condition of their
discrete integrability.
