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Let us begin by recalling a non-integrability criterion for 2-degrees of freedom Hamiltonian systems, with homogeneous potential of integer degree, obtained by Yoshida in 1987 . Such a result is expressed as follows:
Theorem 1: Let
be a homogeneous potential function of degree
and compute the quantity (integrability coefficient)
where is the Hessian matrix of and is a solution of the algebraic equation .
If is in the so-called non-integrability regions Sk, then the 2-degrees of freedom Hamitonian system is non-integrable, i.e. there cannot exist an additional integral which is complex analytic in .
For our purposes, it suffices to consider
and the Sk given by
Now, let us consider the Hamiltonian of the truncated zonal satellite
in Cartesian canonical variables
is the disturbing potential, Pk(x) is the Legendre polynomial of order k, and are coefficients which can be considered as small parameters.
If we carry out a change to cylindrical variables the
Hamiltonian (3) is transformed into
where and the vertical component of the angular momentum, is a first integral since the coordinate is ignorable.
The Hamiltonian (5) has the integrals H= const. and const., which are independent and are in involution (i.e. the Poisson bracket ). Let us suppose that there exists a third first integral F, independent of the other two and in involution with them, that is, verifying since always holds provided F is a first integral.
it is obvious that F does not depend on .
If we perform the reduction of order of the Hamiltonian (5),
by considering as
a parameter, it is clear that if (5) is completely
integrable with integrals H,
and F, the reduction will also be so with integrals
Likewise, if the integrals of (5) are meromorphic,
so are those of the reduction. Moreover, we can assume that F is
since otherwise it suffices to multiply F by a suitable power of
is a first integral
and F are so. In consequence, if (3)
is Liouville integrable, so is
which is obtained by making in the reduction of (5) to two degrees of freedom.
Notice that, as the potential of this 2-degrees of freedom Hamiltonian
consists of a finite number of homogeneous terms, whose degrees vary from
-1 (corresponding to the Keplerian term) to -n-1, we are in conditions
to apply the Yoshida theorem (, theorem
4.1), which allows us to establish the non-existence of an additional meromorphic
integral if the integrability coeficient
of either the lowest or highest order part is in their corresponding non-integrability
regions. To this end, proceeding as Yoshida ,
by performing a suitable change of scale, the Hamiltonian (6)
which is taken as Hamiltonian of an auxiliary problem (see  for details).
As the potential Vn is homogeneous of order m=-n-1,
according to Theorem 1, the non-existence of any other meromorphic
integral simply depends on finding a solution
of the algebraic equation
and on the value of the integrability coefficient
It is easily checked that the system (8)
admits a solution of the form ,
where z0 is a solution to the equation
On the other hand, by using well known properties of Legengre polynomials,
straightforward calculations allow us to compute the trace
to find that the integrability coefficient
As the non integrability regions S-n-1 defined in (2) contain the interval , we conclude that the auxiliary Hamiltonian (7) is non-integrable.
Now, coming back to the Hamiltonian (6), according to Yoshida (, theorem 4.1), in our case, it holds for the lowest order -n-1 and hence (6) is non-integrable. In consequence, as explained before, the original problem (5) is not Liouville integrable through meromorphic integrals.
Let us remark that the choice of the solution to (8) carried out here has allowed us to prove the non-integrability of any truncation of the zonal satellite problem irrespective of whether it ends in even or in odd harmonics, while the solution chosen by Irigoyen and Simó  to set up the non-integrability of the J2-problem (whose truncation ends in J2) would only be useful to prove the non-integrability of truncations ending in even harmonics.