TITLE:
Kovalevskaya, Liapounov, Painleve, Ziglin and the Differential Galois Theory

AUTHOR:
Juan J. Morales-Ruiz
Departament de Matematica Aplicada II
Universitat Politecnica de Catalunya
Pau Gargallo 5, E-08028 Barcelona, Spain
E-mail: morales@ma2.upc.es

INTRODUCTION:
At the light of several recent applications of the differential
Galois theory, in this review I will try make some remarks  of
some important concepts connected with integrability of complex
analytical dynamical systems  started in the seminal works of
Kowalevskaya, Liapounov, Painlev\'e, Picard and Vessiot at the end
of the XIX century. More precisely, our objective is to study the
deep connection between ``integrability" and ``singularity theory"
of complex analytical dynamical systems. The revival of interest
in these problems in the last two decades  was apparently
motivated by the Ziglin and Adler - Van Moerbeke works in the
eighties on integrability of Hamiltonian systems, and by  the
discovery in the sixties of the inverse spectral method for
solving ``integrable" non-linear partial differential equations by
Gardner, Green Kruskal and Miura  and its connection with the
singularity theory: Painlev\'e property, isomodromy deformations,
etc... An idea of the amount of papers published in this field is
given by the 52 pages  of references quoted in the monograph
\cite{AC}.

Although along of this paper essentially there are  no new
results, in section 4 I state a new result. It is a generalization
of the so-called Morales-Ramis theorem about the non-integrability
of an analytical Hamiltonian system by means of the variational
equations. This was conjectured by the author in \cite{MOR6}.

As we will see it is very important to precise carefully the
definition of integrability and also of the Painlev\'e property.
The problem here is involved by the fact that a given differential
equation can ``live" in several different complex analytical
manifolds and, of course, the concept of integrability depends on
this manifold in a very sensible way. Our guiding idea will be
that, as happens with other fields in mathematics, integrability
should be defined in a intrinsic geometrical invariant form. The
last section is devoted to some conclusions about that, based on
previous sections, as well as to some related conjectures.