Characteristically nilpotent Lie groups with flat coadjoint orbits

We study the existence of certain characteristically nilpotent Lie algebras with flat coadjoint orbits. Their connected, simply connected Lie groups admit square-integrable representations modulo the center. There are many examples of nilpotent Lie groups admitting families of dilations and square-integrable representations, but so far no examples admitting square-integrable representations for which the quotient by the center does not admit a family of dilations. In this paper we construct a two-parameter family of characteristically nilpotent Lie groups G(α,β) in dimension $11$, admitting square-integrable representations modulo the center Z, such that G(α,β)/Z does not admit a family of dilations.