1) Put the lines

\institute{Institut Galil\'ee \and Open  University}

\authorrunning{M. Deneufch\^atel, G. H. E. Duchamp, Hoang Ngoc Minh 
\and Allan I. Solomon}

2) Put \mref{diff_eq3} instead of \mref{LES} and a label at the right place

3) \subsection{Through the looking glass...} -->
\section{Through the looking glass...}

4) Condition PI was "primitive independance" now under label \label{prim_ind}

5) Provided definitions in the beginnig of "Through the looking glass" (partial degrees, filtrations).
6) 
[between the Differential Galois group of a differential equation of type \mref{LSE} (acting on the right) ]-->
[between the differential Galois group (acting on the right) of a differential equation of type \mref{diff_eq3}]

7) Added a remark before the conclusion

8) Will call Minh for Allan's [3]