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of the Truncated

In this section we slightly generalize some results previously obtained
by the authors [6] in the sense that we
succeed in proving that the (*J*_{2}
+ *J*_{22})-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

(4) |

with

and ,
are constants which are proportional to the small parameters *J*_{2},
*J*_{22}, 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*=*p _{z}*=0
always remains in the plane

In polar coordinates such a Hamiltonian *H*_{0} is expressed
as

(5) |

It is immediate that if (4) has a first
integral which is analytic in *z*=0, *p _{z}*=0 and is
independent from

(6) |

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,
that is

(7) |

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
to .
Therefore,

This coefficient lies in one of the non-integrability regions given in [6], 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*=*p _{z}*=0.
This result generalizes those previously obtained by the authors ([2],
[6]), where only the non-existence of additional
rational integrals was proved, either for the case
or for the general case.