  We analyze the process of planetary scattering around M0-type stars. To do
this, we carry out N-body simulations with three Jupiter-mass planets close to
their instability limit together with an outer planetesimal disk. This paper
focuses on the analysis of systems in which a single Jupiter-mass planet
survives after the dynamical instability event. The small body reservoirs show
different dynamical behaviors. In fact, our simulations produce particles on
prograde and retrograde orbits, as well as particles whose orbital plane flips
from prograde to retrograde and back again along their evolution. Such
particles are called "Type-F particles". We find strong correlations between
the inclination $i$ and the ascending node longitude $\Omega$ of such
particles. First, $\Omega$ librates around 90$^{\circ}$ or/and 270$^{\circ}$.
This property is very important since it represents a necessary and sufficient
condition for the flipping of an orbit. Moreover, the libration periods of $i$
and $\Omega$ are equal and they are out to phase by a quarter period. We also
remark that the larger the libration amplitude of $i$, the larger the libration
amplitude of $\Omega$. Finally, we analyze the initial conditions of Type-F
particles of all our simulations immediately after the dynamical instability
event, when a single Jupiter-mass planet survives in the system. We carry out
this study with the goal to determine the parameter space that lead to the
flipping of an orbit. Our results suggest that the orbit of a test particle can
flip for any value of its initial eccentricity, although we found only two
Type-F particles with initial inclinations $i <$ 17$^{\circ}$. Moreover, our
study indicates that the minimum value of the inclination of the Type-F
particles in a given system decreases with an increase in the eccentricity of
the giant planet.
