  We observe that the fully nonlinear evolution equations of Rosenau and
Hymann, often abbreviated as $K(n,\,m)$ equations, can be reduced to
Hamiltonian form only on a zero-energy hypersurface belonging to some potential
function associated with the equations. We treat the resulting Hamiltonian
equations by the dynamical systems theory and present a phase-space analysis of
their stable points. The results of our study demonstrate that the equations
can, in general, support both compacton and soliton solutions. For the
$K(2,\,2)$ and $K(3,\,3)$ cases one type of solutions can be obtained from the
other by continuously varying a parameter of the equations. This is not true
for the $K(3,\,2)$ equation for which the parameter can take only negative
values. The $K(2,\,3)$ equation does not have any stable point and, in the
language of mechanics, represents a particle moving with constant acceleration.
