As is seen in Fig.
, the old galactic neutron stars fill a thorus-like
region extended up tens kiloparsecs above the galactic plane with
sharp deficiency of stars toward the galactic axis. As we already
mention earlier, this is a
consequence of the shape of the chosen galactic potential and (in a somewhat
less degree) of the calculated 
-distribution defined by initial
conditions. Unexpectedly
enough, the maximum density lobe is centered close to a distance r from
the galactic center of 
kpc, that is the Sun turns out
inside it. Then it is intriguing to see what the well known tests
of space distribution give in this case. In Fig. we show results
of applying the 
test and
and 
criteria to the obtained distribution
for three specific positions
of the observer with respect to general topology of the distribution
(close to the axis, inside the thorus and outside it),
which describe all different situations. It is clear that a slight change
of the parameters of the model will not distort dramatically the
qualitative picture shown in Fig..
If such a thorus-like distribution actually exists, it will manifest
itself in this manner.
Now return to the crucial question about the applicability of the ergodicity principle to the dynamicas of stars in a real galaxy. This question is directly connected with the question of existence of so-called third isolating integral of motion (see e.g. Binney and Tremaine 1987). If it exists and is a simple-meaning function of space coordinates, the ergodicity is not applicable. Otherwise, if the third integral is not an isolating one or does not exist at all, the situation is more complicated and in any case the strong proof of the ergodicity is required. It is proved at present that the ergodicity is not applicable for a subclass of power-law potentials, but the potential of our Galaxy does not enter into this class (see Sinaj 1982). Numerical calculations also cannot prove strongly the ergodicity due to lack of knowledge of the precise parameters of the potential. From the other side, taking into account of weak collisions (which always should be present in the real galaxy) can support the applicability of the ergodic principle if they are not strong enough to significantly change the equilibrium distribution, but are still sufficient to mix close trajectories in the phase space. After all pro and contra the considerations based on the ergodic hypothesis seem to be quit permissible and in a sence alternative to the methods based on direct integration of equations of motion.