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In this article, we develop a geometric framework to study the notion ofsemi-minimality for the generic type of a smooth autonomous differentialequation $(X,v)$, based on the study of rational factors of $(X,v)$ and ofalgebraic foliations on $X$, invariant under the Lie-derivative of the vectorfield $v$.  We then illustrate the effectiveness of these methods by showing that certainautonomous algebraic differential equation of order three defined over thefield of real numbers --- more precisely, those associated to mixing, compact,Anosov flows of dimension three --- are generically disintegrated.

Rational factors, invariant foliations and algebraic disintegration of compact mixing Anosov flow of dimension $3$