Finding polynomial solutions of linear differential equations is a building block implemented in several algorithms of computer algebra systems. In particular, this is a necessary substep when looking for rational, algebraic or Liouvillian solutions of linear differential equations. When there are no parameters, several algorithms are available, but the general case with parameters is undecidable. However, special families can be handled by ad hoc methods. Such methods were developed by Boucher who applied them to the nice example of integrability of the 3body problem. The key idea there is to rely on a recent result of MoralesRuiz and Ramis who relate complete integrability and differential Galois group. It turns out that special properties of this group can be related to computable properties of an appropriate linear differential equation, which leads Boucher to a ``simple'' sufficient condition for noncomplete integrability.
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æ ç ç è 

+...+ 

+P(a_{1},...,a_{m}) 
ö ÷ ÷ ø 
y(x)=0 
w= 

dp_{i}Ù dq_{i}, 

=X_{H}(g). 
{F,H}:= 


=0, 
H(p,q)= 



 


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.This document was translated from L^{A}T_{E}X by H^{E}V^{E}A.