  This paper studies the dynamics of a family of hamiltonian systems that
originate from Friedman-Lema\^itre-Robertson-Walker space-times with a coupled
field and non-zero curvature. In four distinct cases, previously considered by
Maciejewski, Przybylska, Stachowiak & Szydowski, it is shown that there are
homoclinic connections to invariant submanifolds and the connections split.
These results imply the non-existence of a real-analytic integral independent
of the hamiltonian.
