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Recently, in, the authors have determined that an element $c$ in $H^2(P, P_v)$ is the Chern class of a twisted isotropic realization $p$: $(M, \sigma)$\rightarrow$(P, \pi, \phi)$ if and only if $D_\Lambda(c)$= $e_\phi$, increasing progress towards determining the constructions of twisted isotropic realizations. This problem is divided into 2 steps:

  1. Classify, up to isomorphism, all transversly integral affine structures $F$
  2. For a fixed transversal integral affine structure \Lambda\rightarrow$P$, determine which elements of $H^2(P, P_\Lambda)$ isomorphic to $H^1(P; C^\infty($v^*$F/A$)$

How completely classified are the transversally affine structures on a sympletic foliation? Any references will be greatly appreciated.

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