Beyond Liouvillian transcendence

To a codimension one foliation F defined by a meromorphic 1-form omega, one may associate a Godbillon-Vey sequence (omega(j)), j = 0, 1, ..., of merom orphic 1-forms omega(j) with omega(0) = omega. The sequence is said to have finite length k if w(k) not equal 0 and w(j) = 0 for j > k. The case k = 0 , 1 or 2 corresponds, respectively, to the case where the foliation F has a dditive, affine or projective transverse structure and k less than or equal to 1 is equivalent to the existence of a Liouvillian first integral. These are the only possible cases where the transverse structures come from an a ction of a Lie group on (C) over bar and a non-trivial model for these foli ations is the Riccati differential equation. We propose to go beyond the Li e group transverse structure by studying the case of general k and, for thi s case, we determine a model differential equation, which generalizes the R iccati equation. We also discuss some other related topics.