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Summary of Yoshida's

In this section we present some results about this problem recently obtained by the authors [2], although the proof has been considerably shortened.

By using cylindrical coordinates ,
the homogeneous Hamiltonian for this problem can be written as

(1) |

where ,
,
are constant and *P _{k}*(

From the fact that
is a first integral of (1) it follows that,
if this Hamiltonian admits a meromorphic integral independent from the
other two and in involution with them, such an integral is analytic with
respect to .
Consequently, the integrability in the Liouville sense of (1)
implies that of the Hamiltonian

(2) |

which is obtained by making in (1). It is immediate that (2) admits a particular solution with . However, since an explicit expression for the associate monodromy matrices which are required by Ziglin's method is not known, it is convenient to introduce a suitable limit problem.

The change of scale defined by

gives a new homogeneous Hamiltonian, which after multiplying by
and taking limits as ,
reduces to

(3) |

and the bar in the new variables has been removed to simplify the notation.

The Hamiltonian (3) is of the form required
by Yoshida's theorem [11], since the potential
*V* is homogeneous of degree *k*=-*n*-1. As is well known,
the Hessian of *V* computed on the vector
quoted in section 2 always has the eigenvalue *k*-1, which in our
case turns out to be -*n*-2. It is easy to establish that this is
precisely the value of ,
so that the integrability coefficient reduces to .
Moreover, due to the symmetry with respect to the *OZ* axis, it is
obvious that

Since *V* verifies Laplace equation, we get

a value which lies in the non-integrability region *S*_{-n-1}
and coincides with that previously obtained by the authors in [2]
but with rather more tedious calculations.

Once we have established that the limit problem (3) does not have a complete system of meromorphic integrals, it follows that neither does the problem (2) in a neighbourhood of .

Now, it is enough to apply the usual continuity argument - see [12] or [3] -, since the non-integrability regions correspond to those for which the trace of the monodromy matrices is >2.

However, notice that such an argument is more complicated than simply checking the aforementioned inequality. In fact, it is necessary to verify that the Riemann surface, the poles and the loops which are used to calculate the monodromy matrices do not suffer significant qualitative changes for any problem close to the limit problem, i.e., in a neighbourhood of after performing the change of scale. Nevertheless, this can be verified in an immediate way.

To this end it is enough to realize that the basic solution can be considered
of the form ,
,
where
is a solution of the differential equation

which admits as a first integral

and that the Riemann surface involved in Ziglin's theorem is defined
by the integral

The monodromy group of the normal variation equation is then built by
taking closed circuits around the branch points corresponding to the roots
of *p*(*w*)=0.

In the limit problem, we can take
and *n*+1 simple poles, roots of *w ^{n}*

Notice that this result includes as a particular case the non-integrability
through meromorphic integrals of the truncated two fixed centres problem
in the asymmetric case. The non-integrability of such a problem in the
symmetric case was first established by Irigoyen [4]
by following a scheme similar to that used for the *J*_{2}-problem
in [3]. We would like to point out that the
starting solution used in Irigoyen's work is not suitable for the study
of odd harmonics.

On the other hand, although in (1) only
the harmonics with negative powers have been included, it is easy to realize
that if we add up a finite number of zonal harmonics of the form *r ^{k}V_{k}*
to (1), the conclusion still holds. Such
terms could be envisaged as coming from the truncation of the development
of a perturbation due to a third body by simply assuming some suitable
conditions (e.g. the Keplerian orbit is circular and equatorial).