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In this section we slightly generalize some results previously obtained by the authors  in the sense that we succeed in proving that the (J2 + J22)-problem is not completely integrable through meromorphic integrals.
Assuming that the primary rotates with constant angular velocity
around the Oz axis and using Cartesian coordinates in the rotating
system Oxyz attached to the primary, the Hamiltonian is of the form
and , are constants which are proportional to the small parameters J2, J22, respectively. Notice that whenever the moments of inertia verify A<B<C, as in the case of the Earth, and are positive.
Since any solution with initial conditions z=pz=0 always remains in the plane z=0, by making z=pz=0 in (4), we obtain a new Hamiltonian H0 whose flow is included in that of (4).
In polar coordinates such a Hamiltonian H0 is expressed
It is immediate that if (4) has a first
integral which is analytic in z=0, pz=0 and is
independent from H, it will give rise to another first integral
which is independent from H0 for the subproblem (5).
On the other hand, if this integral was analytic in ,
at least in ,
it would also provide an integral of the auxiliary problem
corresponding to a non-rotating primary.
The problem (6) can be treated in a very
similar way to that of the zonal harmonics. A suitable change of scale
allows us to go to a limit problem, in which the Keplerian term disappears,
The integrability coefficient in polar coordinates is extremely easy
to calculate. According to what is indicated in section 2, it is enough
to consider the solution
This coefficient lies in one of the non-integrability regions given in , so that (7) does not have any additional meromorphic integral.
The usual continuity argument proves that (6) also verifies this property. This allows us to conclude that the original problem (4) does not have any global first integral which is meromorphic and also analytic in w=0, and in z=pz=0. This result generalizes those previously obtained by the authors (, ), where only the non-existence of additional rational integrals was proved, either for the case or for the general case.