  We study the non-trivial central extensions $CEHeis$ of the Heisenberg
algebra $Heis$ recently constructed in {AccBouCE}. We prove that a real form of
$CEHeis$ is one the fifteen classified real four--dimensional solvable Lie
algebras. We also show that $CEHeis$ can be realized (i) as a sub--Lie--algebra
of the Schroedinger algebra and (ii) in terms of two independent copies of the
canonical commutation relations (CCR). This gives a natural family of unitary
representations of $CEHeis$ and allows an explicit determination of the
associated group by exponentiation. In contrast with $Heis$, the group law for
$CEHeis$ is given by nonlinear (quadratic) functions of the coordinates.
