  This paper is a continuation of our article (European J. Math.,
https://doi.org/10.1007/s40879-020-00419-8). The notion of a poor complex
compact manifold was introduced there and the group $Aut(X)$ for a $P^1$-bundle
over such a manifold was proven to be very Jordan. We call a group $G$ very
Jordan if it contains a normal abelian subgroup $G_0$ such that the orders of
finite subgroups of the quotient $G/G_0$ are bounded by a constant depending on
$G$ only.
  In this paper we provide explicit examples of infinite families of poor
manifolds of any complex dimension, namely simple tori of algebraic dimension
zero. Then we consider a non-trivial holomorphic $P^1$-bundle $(X,p,Y)$ over a
non-uniruled complex compact Kaehler manifold $Y$. We prove that $Aut(X)$ is
very Jordan provided some additional conditions on the set of sections of $p$
are met. Applications to $P^1$-bundles over non-algebraic complex tori are
given.
