  Mixing (of all orders) rank-one actions $T$ of Heisenberg group $H_3(\Bbb R)$
are constructed. The restriction of $T$ to the center of $H_3(\Bbb R)$ is
simple and commutes only with $T$. Mixing Poisson and mixing Gaussian actions
of $H_3(\Bbb R)$ are also constructed. A rigid weakly mixing rank-one action
$T$ is constructed such that the restriction of $T$ to the center of $H_3(\Bbb
R)$ is not isomorphic to its inverse.
