  We consider systems of ordinary differential equations (ODEs) of the form
${\cal B}{\mathbf K}=0$, where $\cal B$ is a Hamiltonian operator of a
completely integrable partial differential equation (PDE) hierarchy, and
${\mathbf K}=(K,L)^T$. Such systems, whilst of quite low order and linear in
the components of $\mathbf K$, may represent higher-order nonlinear systems if
we make a choice of $\mathbf K$ in terms of the coefficient functions of $\cal
B$. Indeed, our original motivation for the study of such systems was their
appearance in the study of Painlev\'e hierarchies, where the question of the
reduction of order is of great importance. However, here we do not consider
such particular cases; instead we study such systems for arbitrary $\mathbf K$,
where they may represent both integrable and nonintegrable systems of ordinary
differential equations. We consider the application of the Prelle-Singer (PS)
method --- a method used to find first integrals --- to such systems in order
to reduce their order. We consider the cases of coupled second order ODEs and
coupled third order ODEs, as well as the special case of a scalar third order
ODE; for the case of coupled third order ODEs, the development of the PS method
presented here is new. We apply the PS method to examples of such systems,
based on dispersive water wave, Ito and Korteweg-de Vries Hamiltonian
structures, and show that first integrals can be obtained. It is important to
remember that the equations in question may represent sequences of systems of
increasing order. We thus see that the PS method is a further technique which
we expect to be useful in our future work.
