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### Open Access Journals

DCDS

We consider the isosceles $3$--body problem
with the third particle having a small mass which eventually tend
to zero. Classical McGehee's blow up is not defined because the
matrix of masses becomes degenerate. Following Elbialy [3]
we perform the blow up using the Euclidean norm in the planar
$3$--body problem. We then complete the phase portrait of the flow
in the collision manifold giving the behavior of some branches of
saddle points missing in [3]. The homothetic orbits within
the fixed energy level then provide the necessary recurrence in
order to build a symbolic dynamics. This is done following ideas
of S. Kaplan [6] for the collinear $3$--body problem.
Here the difficulty is the larger number of critical points.

DCDS

In this paper we analyze the non-integrability of the Wilbeforce
spring-pendulum by means of Morales-Ramis theory in where is enough to prove
that the Galois group of the variational equation is not virtually
abelian. We obtain these non-integrability results due to the
algebrization of the variational equation falls into a Heun
differential equation with four singularities and then we apply
Kovacic's algorithm to determine its non-integrability.

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